# Calculus: Single Variable Part 4 โ Applications Coursera Quiz Answers 2022 | All Weeks Assessment Answers [๐ฏCorrect Answer]

Hello Peers, Today we are going to share all week’s assessment and quiz answers of the Calculus: Single Variable Part 4 โ Applications course launched by Coursera totally free of costโโโ. This is a certification course for every interested student.

In case you didn’t find this course for free, then you can apply for financial ads to get this course for totally free.

Coursera, India’s biggest learning platform launched millions of free courses for students daily. These courses are from various recognized universities, where industry experts and professors teach in a very well manner and in a more understandable way.

Here, you will find Calculus: Single Variable Part 4 โ Applications Exam Answers in Bold Color which are given below.

These answers are updated recently and are 100% correctโ answers of all week, assessment, and final exam answers of Calculus: Single Variable Part 4 โ Applications from Coursera Free Certification Course.

Use โCtrl+Fโ To Find Any Questions Answer. & For Mobile User, You Just Need To Click On Three dots In Your Browser & You Will Get A โFindโ Option There. Use These Option to Get Any Random Questions Answer.

## About Calculus: Single Variable Part 4 โ Applications Course

This quick course covers the main ideas of Calculus with one variable, with a focus on understanding the ideas and how to use them. This course is perfect for students who are just starting out in engineering, the physical sciences, or the social sciences.

Course Apply Link – Calculus: Single Variable Part 4 โ Applications

## Calculus: Single Variable Part 4 โ Applications Quiz Answers

### Week 01: Calculus: Single Variable Part 4 โ Applications Coursera Quiz Answers

#### Quiz 01 : Core Homework: Simple Areas

Q1. What is the area between the curve f(x) = \sin^3 xf(x)=sin3x and the xx-axis from x=0x=0 to \displaystyle x=\frac{\pi}{3}x=3ฯโ ?

• \displaystyle \frac{1}{24}241โ
• \displaystyle \frac{23 โ 9\sqrt{3}}{24}2423โ93โโ
• \displaystyle \frac{5}{24}245โ
• \displaystyle \frac{29}{24}2429โ
• \displaystyle \frac{-25 + 15\sqrt{3}}{24}24โ25+153โโ
• \displaystyle \frac{19}{24}2419โ

Q2. Find the area of the bounded region enclosed by the curves y = \sqrt{x}y=xโ and y = x^2y=x2.

Hint: start by drawing the curves in the plane and identifying the appropriate region.

• -3โ3
• \displaystyle \frac{1}{3}31โ
• 11
• -1โ1
• -\displaystyle \frac{1}{3}โ31โ
• 33

Q3. What is the area between the curve y = \sin xy=sinx and the xx-axis for 0\le x\le \pi0โคxโคฯ ?

• \piฯ
• -1โ1
• -\piโฯ
• 22
• -2โ2
• 11

Q4. What is the area between the curve y = \sin xy=sinx and the xx-axis for 0\le x \le 2\pi0โคxโค2ฯ ?

• 00
• 22
• -2โ2
• 11
• 44
• -4โ4

Q5. Recall from the Lecture that the Gini index is defined as

G(f) = \frac{\text{area between } y=x \text{ and } y=f}{\text{area between } y=x \text{ and } y=0} = 2 \int_{x=0}^1 \left(x โ f(x)\right)\, dxG(f)=area between y=x and y=0area between y=x and y=fโ=2โซx=01โ(xโf(x))dx

where f(x)f(x) is the fraction of total income earned by the lowest xx fraction of the populace.

Calculate the Gini index of a country where

f(x) = \frac{2}{5} x^2 + \frac{3}{5} x^3f(x)=52โx2+53โx3

• \displaystyle G(f) = \frac{13}{60}G(f)=6013โ
• \displaystyle G(f) = \frac{13}{30}G(f)=3013โ
• \displaystyle G(f) = \frac{5}{6}G(f)=65โ
• \displaystyle G(f) = \frac{5}{12}G(f)=125โ
• \displaystyle G(f) = \frac{5}{3}G(f)=35โ
• \displaystyle G(f) = \frac{13}{15}G(f)=1513โ

#### Quiz02 : Core Homework: Complex Areas

Q1. Find the area enclosed by the curves y = 1y=1, x = 1x=1, and y = \ln xy=lnx.

Hint: draw the three curves first and identify the region that they enclose. It should look like a right triangle but with a curved hypotenuse.

• \displaystyle e โ \frac{3}{2}eโ23โ
• \ln 2ln2
• e-1eโ1
• 11
• e-2eโ2
• ee

Q2 .Find the area of the bounded region enclosed by the xx-axis, the lines x=1x=1 and x=2x=2 and the hyperbola xy = 1xy=1.

• \displaystyle -\frac{1}{2}โ21โ
• \displaystyle \frac{1}{2}21โ
• \ln 2ln2
• 22
• \ln 3ln3
• 11

Q3. Compute the area in the bounded โ that is, finiteโ regions between y=x(x-1)(x-2)y=x(xโ1)(xโ2) and the xx-axis.

• 11
• 22
• \displaystyle \frac{3}{4}43โ
• \displaystyle \frac{1}{2}21โ
• 00
• \displaystyle \frac{1}{4}41โ

Q4. Find the area of the sector of a circular disc of radius rr (centered at the origin) given by 1 \leq \theta \leq 31โคฮธโค3 (as usual, \thetaฮธ is in radians).

• \displaystyle \frac{\pi r^2}{2}2ฯr2โ
• \displaystyle \frac{2}{3}r^332โr3
• 2r^22r2
• 2r2r
• 2 \pi r^22ฯr2
• r^2r2

Q5. Compute the area enclosed by the cardioid in the figure below. This curve is described by the polar equation r = 1 + \cos\thetar=1+cosฮธ

.

• \displaystyle \frac{5\pi}{2}25ฯโ
• 2\pi2ฯ
• 3\pi3ฯ
• \displaystyle \frac{3\pi}{2}23ฯโ
• \piฯ
• \displaystyle \frac{\pi}{2}2ฯโ

#### Quiz 03: Core Homework: Simple Volumes

Q1. Find the volume of the following solid: for 1 \le x \lt +\infty1โคx<+โ, the intersection of this solid with the plane perpendicular to the xx-axis is a circular disc of radius e^{-x}eโx. Choose โ+\infty+โโ if the resulting integral diverges.

• \displaystyle \frac{\pi-e}{3}3ฯโeโ
• \piฯ
• 1515
• +\infty+โ
• \displaystyle \frac{e^2}{2}2e2โ
• \displaystyle \frac{\pi}{2e^2}2e2ฯโ

Q2. The base of a solid is given by the region lying between the yy-axis, the parabola y=x^2y=x2, and the line y=16y=16 in the first quadrant. Its cross-sections perpendicular to the yy-axis are equilateral triangles. Find the volume of this solid.

• 64\sqrt{3}643โ
• 32\sqrt{3}323โ
• 16\sqrt{3}163โ
• 11
• 2
• 2\sqrt{3}23โ

Q3. The base of a solid is given by the region lying between the yy-axis, the parabola y=x^2y=x2, and the line y=4y=4. Its cross-sections perpendicular to the yy-axis are squares. Find the volume of this solid.

• 22
• \displaystyle\frac{8}{3}38โ
• 44
• 1616
• \displaystyle\frac{16}{3}316โ
• 88

Q4. Find the volume of the solid whose base is the region enclosed by the curve y=\sin xy=sinx and the xx-axis from x=0x=0 to x=\pix=ฯ and whose cross-sections perpendicular to the xx-axis are semicircles.

• \piฯ
• \displaystyle \frac{\pi^2}{16}16ฯ2โ
• \pi^2ฯ2
• \displaystyle \frac{\pi^2}{4}4ฯ2โ
• 00
• \displaystyle \frac{\pi^2}{8}8ฯ2โ

Q5. Consider a cone of height hh over a circular base of radius rr. We computed the volume by slicing parallel to the base. What happens if instead we slice orthogonal to the base? What is the volume element obtained by taking a wedge at angle \thetaฮธ of thickness d\thetadฮธ ?

Hint: if you like, check to see that integrating over 0\le \theta\le 2\pi0โคฮธโค2ฯ gives the correct volume of \pi r^2 h / 3 ฯr2h/3.

• dV = \displaystyle \frac{\pi}{3}r^2hdV=3ฯโr2h

#### Quiz 04:Core Homework: Complex Volumes

Q1. Let DD be the region bounded by the curve y = x^3y=x3, the xx-axis, the line x = 0x=0 and the line x = 2x=2. Find the volume of the region obtained by revolving DD about the xx-axis.

• \displaystyle \frac{128}{7} \pi7128โฯ
• 4 \pi4ฯ
• \displaystyle \frac{64}{7} \pi764โฯ
• 2\pi2ฯ
• None of these
• \displaystyle \frac{64}{4} \pi464โฯ

Q2. Let RR be the region between the curve y = -(x-2)^2+1y=โ(xโ2)2+1 and the xx-axis. Find the volume of the region obtained by revolving RR about the yy-axis.

• \displaystyle \frac{32}{5} \pi532โฯ
• 8 \pi^28ฯ2
• \displaystyle \frac{52}{3} \pi352โฯ
• \displaystyle \frac{16}{3} \pi316โฯ
• \displaystyle \frac{16}{3} \pi^2316โฯ2
• \displaystyle \frac{4}{5} \pi54โฯ

Q3. Find the volume obtained by revolving the region between the curves y = x^3y=x3 and y = \sqrt[3]{x}y=3xโ in the first quadrant about the xx-axis.

• \displaystyle \frac{9}{35} \pi359โฯ
• \displaystyle \frac{16}{35} \pi3516โฯ
• \displaystyle \frac{1}{11} \pi111โฯ
• \displaystyle \frac{26}{35} \pi3526โฯ
• \displaystyle \frac{8}{35} \pi358โฯ
• \displaystyle \frac{32}{35} \pi3532โฯ

Q4. Let DD be the region under the curve y = \ln \sqrt{x}y=lnxโ and above the xx-axis from x = 1x=1 to x = ex=e. Find the volume of the region obtained by revolving DD about the xx-axis.

• \pi(e-1)ฯ(eโ1)
• \displaystyle \frac{\pi(e-1)}{2}2ฯ(eโ1)โ
• \displaystyle \frac{\pi(e-2)}{4}4ฯ(eโ2)โ
• \pi(e-2)ฯ(eโ2)
• \displaystyle \frac{\pi(e-1)}{4}4ฯ(eโ1)โ
• \displaystyle \frac{\pi(e-2)}{2}2ฯ(eโ2)โ

Q5. Let DD be the region from Question 1. What is the volume of the region formed by rotating DD about the line x = 3x=3?

• 24 \pi24ฯ
• \displaystyle \frac{264}{5} \pi5264โฯ
• \displaystyle \frac{184}{3} \pi3184โฯ
• 48 \pi48ฯ
• \displaystyle \frac{56}{5} \pi556โฯ
• \displaystyle \frac{216}{5} \pi5216โฯ

Q6. Let DD be the region bounded by the graph of y = 1-x^4y=1โx4, the xx-axis and the yy-axis in the first quadrant. Which of the following integrals can be used to compute the volume of the region obtained by revolving DD around the line x=5x=5?

• \displaystyle\int_{x=0}^1 2\pi (5-x)(5-x^4) \, dxโซx=01โ2ฯ(5โx)(5โx4)dx
• \displaystyle \int_{x=0}^1 \pi (1-x^4)^2 \, dxโซx=01โฯ(1โx4)2dx
• \displaystyle \int_{y=1}^15 \pi y\sqrt[3]{y-1} \, dyโซy=11โ5ฯy3yโ1โdy
• \displaystyle \int_{x=0}^1 2\pi x(x^4-5) \, dxโซx=01โ2ฯx(x4โ5)dx
• \displaystyle \int_{x=0}^1 2 \pi (5-x)(1-x^4) \, dxโซx=01โ2ฯ(5โx)(1โx4)dx
• \displaystyle \int_{x=0}^1 \pi x^2 (1-x^4) \, dxโซx=01โฯx2(1โx4)dx

### Week 02: Calculus: Single Variable Part 4 โ Applications Coursera Quiz Answers

#### Quiz 01 :Core Homework: Volume and Dimension

Q1. Consider a four-dimensional box (or โrectangular prismโ) with side-lengths 11, 1/21/2, 1/31/3, and 1/41/4. What is the 44-dimensional volume of this box?

• \displaystyle \frac{1}{24}241โ
• \displaystyle \frac{1}{12}121โ
• \displaystyle 11
• \displaystyle \frac{1}{2}21โ
• \displaystyle \frac{1}{6}61โ

Q2. In the 44-d box of Question 1, what is the โdiameterโ โi.e., the farthest distance between two points in the box?

Hint: think in terms of diagonals.

• \displaystyle \frac{5}{2\sqrt{3}}23โ5โ
• \displaystyle \frac{1}{2\sqrt{6}}26โ1โ
• \displaystyle \frac{\sqrt{205}}{12}12205โโ
• 22
• \displaystyle \frac{25}{12}1225โ

Q3. High-dimensional objects are everywhere and all about. Letโs consider a very simple model of the space of digital images. Assume a planar digital image (such as that captured by a digital camera), where each pixel is given values that encode color and intensity of light. Letโs assume that this is done via an RGB (red/blue/green) model. Though there are many RGB model specifications, let us use one well-suited for mathematics: to each pixel on associates three numbers (R,G,B)(R,G,B), each taking a value in [0,1][0,1].

Since the red/blue/green values are independent, each pixel has associated to it a 3-d cube of possible color values. Consider a (fairly standard) 10-megapixel camera. If I were to consider the โspace of all imagesโ that my camera can capture, what does the space look like?

Note: thereโs no calculus in this problemโฆjust counting!

• A unit ball of dimension 3\times 10^63ร106
• A unit cube of dimension 3\times 10^63ร106
• A unit simplex of dimension 3\times 10^63ร106
• A unit cube of dimension 3\times10^73ร107
• A unit cube of dimension 3\times 10^{10}3ร1010

#### Quiz 02 :Core Homework: Arclength

Q1. Find the arc length of the curve \displaystyle y = \left( x + \frac{5}{9} \right)^{3/2}y=(x+95โ)3/2 from x = 0x=0 to x = 3x=3.

• 11
• 88
• 77
• \displaystyle \frac{63}{4}463โ
• \displaystyle \frac{3}{2}23โ
• \displaystyle \frac{21}{2}221โ

Q2. Find the arc length of the curve y = -\ln (\cos x)y=โln(cosx) from x = 0x=0 to x = \displaystyle \frac{\pi}{4}x=4ฯโ.

Hint: you may need to use that

\int \sec x\, dx = \ln \left|\sec x + \tan x \right| + Cโซsecxdx=lnโฃsecx+tanxโฃ+C

• \ln \sqrt2ln2โ
• \ln (\sqrt{2} โ 1)ln(2โโ1)
• \ln (\sqrt2 + 1)ln(2โ+1)
• \displaystyle \ln \frac{\sqrt2}{2}ln22โโ
• \displaystyle\ln \left( \frac{\sqrt2}{2} + 1 \right)ln(22โโ+1)
• 11

Q3. The so-called cuspidal cubic is given parametrically by the equations

x = t^3, \quad y = t^2x=t3,y=t2

Compute the arc length of this curve as tt goes from -1โ1 to 11. Provide a numeric answer rounded to two decimal places.

Q4. Consider the spiral given by the parametric equations

x = t^{-k} \cos t, \quad y = t^{-k} \sin tx=tโkcost,y=tโksint

where k > 0k>0. Denote by L_kLkโ its arc length as tt moves from 2\pi2ฯ to +\infty+โ. Which of the following statements are true? Select all that apply.

Hint: in Lecture we studied the case k=1k=1: see the figure from the Lecture if you need help visualizingโฆ

• L_kLkโ is finite for k \gt 1k>1, and infinite for k \leq 1kโค1
• \displaystyle L_k = \int_{t=2\pi}^{+\infty} \frac{\sqrt{k^2 + t^2}}{t^{k+1}} \, dtLkโ=โซt=2ฯ+โโtk+1k2+t2โโdt
• \displaystyle L_k = \int_{t=2\pi}^{+\infty} \frac{\sqrt{1 + k^2 t^2}}{t^{k+1}} \, dtLkโ=โซt=2ฯ+โโtk+11+k2t2โโdt
• L_kLkโ is finite for k \lt 1k<1, and infinite for k \geq 1kโฅ1
• \displaystyle L_k = \int_{t=2\pi}^{+\infty} \frac{\sqrt{1 + t^2}}{k t^{k+1}} \, dtLkโ=โซt=2ฯ+โโktk+11+t2โโdt
• \displaystyle L_k = \int_{t=2\pi}^{+\infty} \frac{\sqrt{1 + t^2}}{t^{k+1}} \, dtLkโ=โซt=2ฯ+โโtk+11+t2โโdt

Q5. At the close of this lecture we saw an example of a fractal โ the so-called Koch snowflake. A similar example is given by the following procedure. Starting with a line segment of length 11 (labelled โ1โ in the figure below), remove the middle third and replace it by a square hat to obtain the curve โ2โ. Perform the same operation on each line segment in โ2โ to obtain โ3โ.

Doing this ad infinitum yields another fractal โ that is, a bounded compact curve of infinite length!

But what is the exact length of the curve obtained after a finite number nn of iterations?

[Images courtesy of Wikimedia Commons]

• \displaystyle \left( \frac{5}{3} \right)^n(35โ)n
• \displaystyle \left( \frac{4}{3} \right)^n(34โ)n
• \displaystyle \left( \frac{3}{4} \right)^n(43โ)n
• \displaystyle \left( \frac{3}{5} \right)^n(53โ)n
• \displaystyle \left( \frac{5}{4} \right)^n(45โ)n
• \displaystyle \left( \frac{4}{5} \right)^n(54โ)n

#### Quiz 03:Core Homework: Surface Area

Q1. Think of the sphere of radius 11 as obtained by revolving the curve y = \sqrt{1-x^2}y=1โx2โ about the xx-axis. For any -1 \leq a \lt b \leq 1โ1โคa<bโค1, calculate the surface area of the slice between x=ax=a and x=bx=b.

• \displaystyle 4\pi \sqrt{\frac{b+a}{2}}4ฯ2b+aโโ
• 2\pi (b^2 + a^2)2ฯ(b2+a2)
• \displaystyle 4\pi \sqrt{\frac{b-a}{2}}4ฯ2bโaโโ
• 2\pi(b-a)2ฯ(bโa)
• 2\pi (b^2 โ a^2)2ฯ(b2โa2)
• 2\pi(b+a)2ฯ(b+a)

Q2. A typical dish antenna is built as a surface of revolution obtained by revolving a parabola about an axis of symmetry. One of the main benefits of this design is that the resulting antenna exhibits very high gains in the direction towards which it points, making it well-suited for applications in which a strong directionality is needed โsuch as TV reception and radar.

We can model such a parabolic antenna as the surface of revolution obtained by revolving the function

y = \sqrt{\frac{K}{4}} x^2, \qquad 0 \leq x \leq Ry=4Kโโx2,0โคxโคR

about the yy-axis. Here RR is the radius of the antenna, and KK โthe curvature at the tipโ controls how flat it is. Compute the surface area of this antenna in terms of the parameters RR and KK.

1 point

• \displaystyle \frac{4\sqrt{2}}{3} R^{3/2}K^{-1/4}342โโR3/2Kโ1/4
• \displaystyle \frac{\pi}{K} \left[ \left( 1 + 2RK \right)^{1/2} โ 1 \right]Kฯโ[(1+2RK)1/2โ1]
• \displaystyle \frac{2\sqrt{2}}{3} R^{1/2} K^{-3/4}322โโR1/2Kโ3/4
• \displaystyle \frac{2\pi}{3K} \left[ \left( 1 + KR^2 \right)^{3/2} โ 1 \right]3K2ฯโ[(1+KR2)3/2โ1]
• \displaystyle \frac{2\pi}{3K} \left[ \left( 1 + 2RK \right)^{3/2} โ 1 \right]3K2ฯโ[(1+2RK)3/2โ1]
• \displaystyle \frac{\pi}{K} \left[ \left( 1 + KR^2 \right)^{1/2} โ 1 \right]Kฯโ[(1+KR2)1/2โ1]

Q3. Consider the truncated circular cone in the figure (just the sides, not including the bottom and top).

It can be modeled as the surface of revolution obtained by revolving the line

x = R_1 + (R_2-R_1)\frac{y}{h}, \qquad 0 \leq y \leq hx=R1โ+(R2โโR1โ)hyโ,0โคyโคh

about the yy-axis. Which of the following expressions describes its surface area in terms of the parameters hh, R_1R1โ and R_2R2โ ?

• \displaystyle \frac{\pi}{2} (R_1 + R_2) \sqrt{h^2 + (R_2-R_1)^2}2ฯโ(R1โ+R2โ)h2+(R2โโR1โ)2โ
• \displaystyle \pi(R_1 + R_2) \left(h^2 + (R_2-R_1)^2\right)^{3/2}ฯ(R1โ+R2โ)(h2+(R2โโR1โ)2)3/2
• \displaystyle \pi(R_1 + R_2) \sqrt{h^2 + (R_2-R_1)^2}ฯ(R1โ+R2โ)h2+(R2โโR1โ)2โ
• \displaystyle \frac{\pi(R_1 + R_2)}{2\sqrt{h^2 + (R_2-R_1)^2}}2h2+(R2โโR1โ)2โฯ(R1โ+R2โ)โ
• \displaystyle \frac{\pi(R_1 + R_2)}{\sqrt{h^2 + (R_2-R_1)^2}}h2+(R2โโR1โ)2โฯ(R1โ+R2โ)โ
• \displaystyle \frac{\pi}{2} (R_1 + R_2) \left(h^2 + (R_2-R_1)^2\right)^{3/2}2ฯโ(R1โ+R2โ)(h2+(R2โโR1โ)2)3/2

Q4. Consider a circular tent whose roof is made of fabric hanging from the rim of the walls of the tent and supported at a central pole.

If you look at the curve that the fabric roof forms along any radial cross-section, you will discover a catenary โthat is, a hyperbolic cosine. Modeling the roof as the surface of revolution obtained by revolving the curve

y = R \cosh\left( 1 โ \frac{x}{R} \right) , \qquad 0 \leq x \leq Ry=Rcosh(1โRxโ),0โคxโคR

around the yy-axis, which of the following integrals computes its surface area?

• \displaystyle 2\pi \int_{u=0}^1 u \cosh(1-u) \, du2ฯโซu=01โucosh(1โu)du
• \displaystyle 2\pi R \int_{u=0}^1 u \cosh(1-u) \, du2ฯRโซu=01โucosh(1โu)du
• \displaystyle 2\pi R^2 \int_{u=0}^1 u \cosh(1-u) \, du2ฯR2โซu=01โucosh(1โu)du
• \displaystyle 2\pi R \int_{u=0}^1 u \sinh(1-u) \, du2ฯRโซu=01โusinh(1โu)du
• \displaystyle 2\pi R^2 \int_{u=0}^1 u \sinh(1-u) \, du2ฯR2โซu=01โusinh(1โu)du
• \displaystyle 2\pi \int_{u=0}^1 u \sinh(1-u) \, du2ฯโซu=01โusinh(1โu)du

### Week 03: Calculus: Single Variable Part 4 โ Applications Coursera Quiz Answers

#### Quiz 01:Core Homework: Work

Q1. How much work is needed to lift a 40 \text{ kg}40 kg television up a height of 22 meters? Take the acceleration of gravity to be g = 10\,\mathrm{m}/\mathrm{s}^2g=10m/s2.

A reminder on units: recall that, in the International System of Units, length is measured in meters (\mathrm{m}m), time in seconds (\mathrm{s}s), and mass in kilograms (\mathrm{kg}kg). The unit of force is called a newton (\mathrm{N}N), and that of work a joule (\mathrm{J}J). Newtonโs Second Law, F = maF=ma, tells us that

1\,\mathrm{N} = 1\,\mathrm{kg}\,\mathrm{m}/\mathrm{s}^21N=1kgm/s2

The basic definition of work as a product of force and distance then yields

1\,\mathrm{J} = 1\,\mathrm{N}\,\mathrm{m} = 1\,\mathrm{kg}\,\mathrm{m}^2/\mathrm{s}^21J=1Nm=1kgm2/s2

• 1,\!000\,\mathrm{J}1,000J
• 80\,\mathrm{J}80J
• 800\,\mathrm{J}800J
• 400\,\mathrm{J}400J
• 1,\!600\,\mathrm{J}1,600J
• 40\,\mathrm{J}40J

Q2. Your swimming pool is 3\,\mathrm{m}3m deep, 10\,\mathrm{m}10m long and 6\,\mathrm{m}6m wide. If the pool is initially full, how much work is required to drain two thirds of the water in the pool (that is, until the water is only 1\,\mathrm{m}1m deep)? Assume that the density of water is 1,\!000\,\mathrm{kg}/\mathrm{m}^31,000kg/m3, and that the acceleration of gravity is g = 10\,\mathrm{m}/\mathrm{s}^2g=10m/s2.

• 1.35\cdot 10^6\,\mathrm{J}1.35โ106J
• 1.25\cdot 10^5\,\mathrm{J}1.25โ105J
• 2.4 \cdot 10^4\,\mathrm{J}2.4โ104J
• 1.2 \cdot 10^6\,\mathrm{J}1.2โ106J
• 1.2\cdot 10^5\,\mathrm{J}1.2โ105J
• 2.7\cdot 10^5\,\mathrm{J}2.7โ105J

Q3. A 100100 meter long cable of linear mass density 0.1\,\mathrm{kg}/\mathrm{m}0.1kg/m hangs over a very high vertical cliff. Assuming that there is no friction, how much work is needed to to lift this cable up to the top of the cliff? Assume that the acceleration due to gravity is g = 10\,\mathrm{m}/\mathrm{s}^2g=10m/s2.

• 500\,\mathrm{J}500J
• 100\,\mathrm{J}100J
• 2,\!500\,\mathrm{J}2,500J
• 10,\!000\,\mathrm{J}10,000J
• 5,\!000\,\mathrm{J}5,000J
• 50\,\mathrm{J}50J

Q4. Assume that a sports carโs acceleration aa increases linearly with its position xx as a(x) = xa(x)=x. Since the car is burning fuel, its mass mm decreases; assume the decrease is exponential in xx as m(x) = 1 + e^{-x}m(x)=1+eโx. How much work is done in driving the car from x=0x=0 to x = 3x=3 ?

Hint: remember Newtonโs Second Law, F=maF=ma. In our case, both mass and acceleration are functions of xx.

• 3e^2 โ 13e2โ1
• \displaystyle 3 + \frac{3}{e^3}3+e33โ
• \displaystyle 1 โ \frac{2}{e}1โe2โ
• \displaystyle \frac{9}{2}+\frac{2}{e^3}29โ+e32โ
• \displaystyle \frac{11}{2}-\frac{4}{e^3}211โโe34โ
• \displaystyle \frac{2}{e

#### Quiz 02:Core Homework: Elements

Q1. Consider a dam of height HH and width WW that has a perfectly vertical face facing the water, which reaches all the way up to the damโs height. If the water has weight density \rhoฯ, what is the total force the water exerts against the face of the dam?

• \displaystyle \frac{1}{4}H^2 W \rho41โH2Wฯ
• \displaystyle \frac{1}{2}H W \rho21โHWฯ
• \displaystyle \frac{1}{4}H W^2 \rho41โHW2ฯ
• H^2 W \rhoH2Wฯ
• \displaystyle \frac{1}{2}H W^2 \rho21โHW2ฯ
• \displaystyle \frac{1}{2}H^2 W \rho21โH2Wฯ

Q2. Consider two potential income streams, each valued based on an assumption of a constant return on investment at rate r>0r>0. The first, I_1I1โ, starts off slow, then peaks, and then decreases. The second, I_2I2โ, starts off high, then decreases. Both oscillate eventually with the same period. The specific formulae are:

I_1(t) = I_0 + A\sin\frac{\pi t}{P} \quad ; \quad I_2(t) = I_0 + A\cos\frac{\pi t}{P}I1โ(t)=I0โ+AsinPฯtโ;I2โ(t)=I0โ+AcosPฯtโ

Here, I_0>0I0โ>0 is a constant (the baseline income), A>0A>0 is a constant (the amplitude of fluctuation) and P>0P>0 is a constant (the half-period). Assume \pi \gt Prฯ>Pr. Which income stream has the greater present value over the time interval [0,P][0,P] ? Which has the greater present value over the time interval [0, +\infty)[0,+โ) ?

Hints: (1) Which constants are important? I_0I0โ? AA? PP? rr? (2) You may want a reduction formula like that from Lecture 22. (3) If you get stuck in the algebra, try using WolframAlpha.

• On [0,P][0,P], PV_1=PV_2PV1โ=PV2โ; but on [0,+\infty)[0,+โ), PV_1\lt PV_2PV1โ<PV2โ.
• PV_1 \lt PV_2PV1โ<PV2โ both on [0, P][0,P] and [0, +\infty)[0,+โ).
• On [0,P][0,P], PV_1=PV_2PV1โ=PV2โ; but on [0,+\infty)[0,+โ), PV_1>PV_2PV1โ>PV2โ.
• On [0,P][0,P], PV_1\lt PV_2PV1โ<PV2โ, but on [0,+\infty)[0,+โ), PV_1\gt PV_2PV1โ>PV2โ.
• PV_1 = PV_2PV1โ=PV2โ both on [0, P][0,P] and [0, +\infty)[0,+โ).
• PV_1 \gt PV_2PV1โ>PV2โ both on [0, P][0,P] and [0, +\infty)[0,+โ).

Q3 .We have learned about present value of an income stream I(t)I(t); one may also reverse the derivation to determine the future value of the income at a time t=Tt=T. The future value element of I(t)I(t) is

dFV = e^{r(T-t)}I(t)dt,dFV=er(Tโt)I(t)dt,

assuming a continuous compounding at fixed interest rate rr.

If you save for a childโs college at a rate of \$5,\!000 / \mathrm{year}$5,000/year starting at the childโs birth, how much money will be available when she is 2020? Assume a fixed 5\%5% return on investments.

• FV = \$100,\!000eFV=$100,000e
• FV = \$50,\!000 eFV=$50,000e
• FV = \$50,\!000\sqrt{e}FV=$50,000eโ
• FV = \$100,\!000(e-1)FV=$100,000(eโ1)
• FV = \$500,\!000FV=$500,000
• FV = \$100,\!000FV=$100,000

Q4. Consider a cantilever beam of length LL. Suppose that NN people, each of mass m_0m0โ, stand on it equally spaced, so that their combined weight is supported uniformly along the beam. If L = 20\,\mathrm{m}L=20m, m_0 = 75\,\mathrm{kg}m0โ=75kg and, at the point of attachment, the beam can withstand a maximum torque of \tau_\mathrm{max} = 1.5\cdot 10^6\,\mathrm{N}\cdot\mathrm{m}ฯmaxโ=1.5โ106Nโm, what is the maximum number of people that can stand on it? Assume the acceleration of gravity to be g = 10\,\mathrm{m}/\mathrm{s}^2g=10m/s2.

• 25 people.
• 50 people.
• 400 people.
• 300 people.
• 200 people.
• 100 people.Suppose that a radiator is turned off at t=0t=0; after that, the amount of heat generated by the radiator is described by the heat flow element

dQ = Q_0 e^{-\lambda t} dtdQ=Q0โeโฮปtdt

Q5. where both Q_0Q0โ and \lambdaฮป are positive constants. What is the total amount of heat radiated from the moment it is turned off?

• \sqrt{\lambda} Q_0ฮปโQ0โ
• \lambda Q_0ฮปQ0โ
• Q_0 e^\lambdaQ0โeฮป
• \displaystyle \frac{Q_0}{\lambda}ฮปQ0โโ
• \lambda^2 Q_0ฮป2Q0โ
• Q_0 e^{-\lambda}Q0โeโฮป

### Week 04: Calculus: Single Variable Part 4 โ Applications Coursera Quiz Answers

#### Quiz 01:Core Homework: Averagez

Q1. Find the average value of \displaystyle f(x) = \frac{1}{\sqrt{4x โ 3}}f(x)=4xโ3โ1โ from x = 3x=3 to x = 21x=21.

• \displaystyle \frac{1}{12}121โ
• \displaystyle -\frac{2}{9}โ92โ
• 33
• \displaystyle \frac{1}{18}181โ
• \displaystyle \frac{3}{2}23โ
• \displaystyle \frac{1}{6}61โ

Q2. Calculate the average of the function f(x) = x^3 \sqrt{1+x^2}f(x)=x31+x2โ over the interval 0 \leq x \leq \sqrt{3}0โคxโค3โ.

• \displaystyle \frac{58}{15}1558โ
• \displaystyle \frac{128}{15}15128โ
• \displaystyle \frac{128}{15\sqrt{3}}153โ128โ
• \displaystyle \frac{2}{15\sqrt{3}} \left( 1 + \sqrt{2} \right)153โ2โ(1+2โ)
• \displaystyle \frac{2}{15} \left( 1 + \sqrt{2} \right)152โ(1+2โ)
• \displaystyle \frac{58}{15\sqrt{3}}153โ58โ

Q3. It is intuitively clear that the average value of xx over a circle of radius 11 (given by the equation x^2 + y^2 = 1x2+y2=1) is zero. But what is the average value of x^2x2 over this circle?

Hint: notice that this is an average over a curve, so you will need to integrate with respect to the arc length element dLdL. In order to make your calculations easier, use the parametrization

x = \cos t, \quad y = \sin t, \qquad 0 \leq t \leq 2\pix=cost,y=sint,0โคtโค2ฯ

• 00
• \displaystyle \frac{1}{2\pi}2ฯ1โ
• \displaystyle \frac{1}{4}41โ
• \displaystyle \frac{1}{2}21โ
• \displaystyle \frac{2}{\pi}ฯ2โ
• \displaystyle \frac{\pi}{4}4ฯโ

Q4. Let us model a mountain as a circular cone of height hh whose base has radius RR. You can see it as the surface obtained by revolving the line

y = h \left( 1 โ \frac{x}{R} \right), \qquad 0 \leq x \leq Ry=h(1โRxโ),0โคxโคR

about the yy-axis. What is the average height of the points on the surface of the mountain?

Hint: This average is an integral with respect to area. You may wish to take as area element an infinitesimal annulus centered at the origin.

• \displaystyle \frac{h}{3}3hโ
• \displaystyle \frac{h}{6}6hโ
• \displaystyle \frac{1}{2} \pi R^2 h21โฯR2h
• \displaystyle \frac{h}{2}2hโ
• \displaystyle \frac{1}{6} \pi R^2 h61โฯR2h
• \displaystyle \frac{1}{3} \pi R^2 h31โฯR2h

Q5. What is the average of (x-1)^2(xโ1)2 over the domain 1\leq \vert x\vert \leq 31โคโฃxโฃโค3. Be careful!

• 11
• \displaystyle\frac{16}{3}316โ
• \displaystyle\frac{4}{3}34โ
• \displaystyle\frac{32}{3}332โ
• \displaystyle\frac{8}{3}38โ
• 88

#### Quiz 02:Core Homework: Centroids

Q1. Find the coordinates (\overline{x},\overline{y})(x,yโ) of the centroid of the region bounded by y=\sin xy=sinx and y=\cos xy=cosx for \displaystyle 0 \leq x \leq \frac{\pi}{4}.0โคxโค4ฯโ.

1 point

• \displaystyle \overline{x}=\frac{\pi\sqrt{2}}{\sqrt{2}-1}x=2โโ1ฯ2โโ,
• \displaystyle \overline{y}=\frac{1}{\sqrt{2}-1}yโ=2โโ11โ
• \displaystyle \overline{x}=\frac{\sqrt 2}{2}x=22โโ,
• \displaystyle \overline{y}=\frac{\sqrt 2}{2}yโ=22โโ
• \displaystyle \overline{x}=\frac{1}{\sqrt 2 -1}x=2โโ11โ,
• \displaystyle \overline{y}=\frac{1}{\sqrt 2-1}yโ=2โโ11โ
• \displaystyle \overline{x}=\frac{\pi}{8}x=8ฯโ,
• \displaystyle \overline{y}=\sqrt \frac{2- \sqrt 2}{2}yโ=22โ2โโโ
• \displaystyle \overline{x}=\frac{\pi\sqrt{2}-4}{4(\sqrt{2}-1)}x=4(2โโ1)ฯ2โโ4โ,
• \displaystyle \overline{y}=\frac{1}{4(\sqrt{2}-1)}yโ=4(2โโ1)1โ
• \displaystyle \overline{x}=\pi\sqrt 2 x=ฯ2โ,
• \displaystyle \overline{y}=1yโ=1

Q2. Find the the coordinates (\overline{x},\overline{y})(x,yโ) of the centroid of the region defined by \displaystyle |x+y| \leq 1โฃx+yโฃโค1, -1\leq x\leq 1โ1โคxโค1, and -1\leq y\leq 1โ1โคyโค1.

Hint 1: draw a picture!

• \displaystyle (\overline{x},\overline{y})=\left(\frac{1}{\sqrt2},\frac{1}{\sqrt 2}\right)(x,yโ)=(2โ1โ,2โ1โ)
• \displaystyle (\overline{x},\overline{y})= \left(\frac{1}{2},\frac{1}{2}\right)(x,yโ)=(21โ,21โ)
• (\overline{x},\overline{y})=\displaystyle \left(-\frac{1}{\sqrt2},\frac{1}{\sqrt 2}\right)(x,yโ)=(โ2โ1โ,2โ1โ)
• (\overline{x},\overline{y})=\displaystyle (0,0)(x,yโ)=(0,0)
• (\overline{x},\overline{y})=\displaystyle (1,1)(x,yโ)=(1,1)
• \displaystyle (\overline{x},\overline{y})= \left(-\frac{1}{\sqrt2}, -\frac{1}{\sqrt 2}\right)(x,yโ)=(โ2โ1โ,โ2โ1โ)

Q3. Compute the center of mass of a thin rod with density \rho(x)=e^{-ax}ฯ(x)=eโax for a\gt 0a>0 a constant and 0\leq x\lt\infty0โคx<โ. (Yes, i know, itโs not-so-physical to talk about infinite rodsโฆtrust me, you will care about this result soon!)

• \displaystyle\overline{x} = e^ax=ea
• \displaystyle\overline{x} = \frac{1}{a}x=a1โ
• \displaystyle\overline{x} = ax=a
• \displaystyle\overline{x} = \frac{1}{a^2}x=a21โ
• \displaystyle\overline{x}x does not exist (the integral diverges).
• \displaystyle\overline{x} = 1x=1

Q4. Find the coordinates (\overline{x},\overline{y})(x,yโ) of the centroid of the union of the following two discs:

D_1: x^2 + y^2 \leq 4 \qquad\text{and}\qquad D_2: (x โ 4)^2 + (y โ 2)^2 \leq 1D1โ:x2+y2โค4andD2โ:(xโ4)2+(yโ2)2โค1

Hint: replace each disc with a vertex at its centroid. What โmassโ should you assign to each vertex?

• \displaystyle (\overline{x},\overline{y})=\left( \frac{4\pi}{5}, \frac{2\pi}{5} \right)(x,yโ)=(54ฯโ,52ฯโ)
• (\overline{x},\overline{y})=(0,0)(x,yโ)=(0,0)
• (\overline{x},\overline{y})=\displaystyle \left( \frac{4}{5}, \frac{2}{5} \right)(x,yโ)=(54โ,52โ)
• \displaystyle (\overline{x},\overline{y})=\left( \frac{4}{\pi}, \frac{2}{\pi} \right)(x,yโ)=(ฯ4โ,ฯ2โ)
• \displaystyle (\overline{x},\overline{y})= (4\pi,2\pi)(x,yโ)=(4ฯ,2ฯ)
• \displaystyle (\overline{x},\overline{y})= \left(2,1\right)(x,yโ)=(2,1)

Q5. Find the the coordinates (\overline{x},\overline{y})(x,yโ) of the center of mass of the region between the xx-axis, the yy-axis, and the lines x=2x=2 and \displaystyle y=x+2y=x+2, with density (mass-per-unit-area) \rho=3xฯ=3x.

Hint: remember, this is a center-of-mass, not a centroid, so youโll need to integrate with respect to dM=\rho\cdot dAdM=ฯโdA.

• (\overline{x},\overline{y})=(1,2)(x,yโ)=(1,2)
• (\overline{x},\overline{y})=\displaystyle \left(\frac{7}{5},\frac{17}{5}\right)(x,yโ)=(57โ,517โ)
• (\overline{x},\overline{y})=\displaystyle \left(\frac{7}{5},\frac{17}{10}\right)(x,yโ)=(57โ,1017โ)
• (\overline{x},\overline{y})=\displaystyle \left(10, \frac{17}{3}\right)(x,yโ)=(10,317โ)
• (\overline{x},\overline{y})=\displaystyle \left(\frac{1}{3},\frac{17}{3}\right)(x,yโ)=(31โ,317โ)
• (\overline{x},\overline{y})=\displaystyle \left(\frac{14}{3},\frac{17}{3}\right)(x,yโ)=(314โ,317โ)

#### Quiz 03:Core Homework: Moments and Gyrations

q1. Three particles, each of mass mm, are located at distances r_1r1โ, r_2r2โ and r_3r3โ respectively from a fixed axis of rotation AA. We now place a fourth particle, also of mass mm, at some distance rr from the axis AA. If the moment of inertia of all four particles is twice as big as the moment of inertia of the first three, what is rr ?

Note: this question doesnโt really use any calculus, but it will give you practice at remembering what moment of inertia means.

• r = ( r_1 + r_2 + r_3 ) \ln 2r=(r1โ+r2โ+r3โ)ln2
• \displaystyle r = \frac{2}{3} ( r_1 + r_2 + r_3 )r=32โ(r1โ+r2โ+r3โ)
• \displaystyle r = \sqrt{r_1^2+r_2^2+r_3^2}r=r12โ+r22โ+r32โโ
• r = \sqrt[3]{r_1 r_2 r_3}r=3r1โr2โr3โโ
• \displaystyle r = \frac{r_1^2}{r_2}+\frac{r_2^2}{r_3}+\frac{r_3^2}{r_1}r=r2โr12โโ+r3โr22โโ+r1โr32โโ
• r = \sqrt[3]{2 \left( r_1^3 + r_2^3 + r_3^3 \right) }r=32(r13โ+r23โ+r33โ)โ

Q2. In mathematics, an annulus is defined as the region between two circles with a common center. Assume you are given an annulus with outer radius RR, inner radius rr, and mass MM distributed uniformly. What is its moment of inertia about the central axis shown in the picture below?

Hint: this problem becomes easier if you watch the bonus lecture first!

• \displaystyle I_\text{annulus} = \frac{1}{4}M(R^2-r^2)Iannulusโ=41โM(R2โr2)
• \displaystyle I_\text{annulus} = M(R-r)\sqrt{R^2-r^2}Iannulusโ=M(Rโr)R2โr2โ
• \displaystyle I_\text{annulus} = \frac{1}{2}M(R^2+r^2)Iannulusโ=21โM(R2+r2)
• \displaystyle I_\text{annulus} = \frac{1}{4}M(R^2+r^2)Iannulusโ=41โM(R2+r2)
• \displaystyle I_\text{annulus} = \frac{1}{2}M(R^2-r^2)Iannulusโ=21โM(R2โr2)
• \displaystyle I_\text{annulus} = \frac{1}{2}MR^2-\frac{1}{4}Mr^2Iannulusโ=21โMR2โ41โMr2

Q3. A hollow cylindrical shell of length LL and radius RR is rotated about the an axis as shown in the picture.

You may assume that this cylindrical shell does not have โcapsโ at either the left or the right edge, and that its mass MM is distributed uniformly along the surface.You may also assume that RR is small enough that the piece of this cylinder at any distance rr from the axis of rotation is a circle. What is its moment of inertia?

HInt: start by computing the area AA and then the density \rho=M/Aฯ=M/A. Then, setting rr to be a radial coordinate (distance-to-axis), the moment-of-inertia element is dI=\rho r^2 dAdI=ฯr2dA. For dAdA, use the approximation implied by the โRR is smallโ assumption.

• \displaystyle \frac{2}{5}M(L^2+\pi R^2)52โM(L2+ฯR2)
• \displaystyle \frac{2\pi}{3}{MLR}32ฯโMLR
• \displaystyle \frac{1}{4}ML^241โML2
• \displaystyle \frac{2}{3}ML^232โML2
• \displaystyle \frac{1}{4} MR^241โMR2
• \displaystyle \frac{1}{3}ML^231โML2

Q4. You need to install a heavy front door in your home. For simplicity, assume that the door has uniform density, has total mass MM, and fills a rectangular entry of height hh and width \ellโ. You have two choices:

1. a single-door, with a single set of hinges on one side; or
2. double-doors, meaning spilt down the middle into two rectangular โhalf-doorsโ of height hh and width \ell/2โ/2, each with hinges on the side.

You would guess that the single-door option is harder to open. How much more is the moment of inertia II of the single door than the (net) II of the two half-doors?

• Twice as much
• Four times as much
• Six times as much
• Itโs the same
• Three times as much
• Four-thirds as much

### week 05: Calculus: Single Variable Part 4 โ Applications Coursera Quiz Answers

#### Quiz 01: Core Homework: Fair Probability

Q1. The result of flipping a single coin is either heads, H, or tails ,T, each one of them with probability 1/21/2 โsuch a coin is said to be fair. If you flip the same coin a second time, there are four possible combinations of the results of both tosses โHH, HT, TH and TTโ, each one of them equally probable. Think of what happens when you do it yet once more: what is the probability of obtaining two heads and one tail, in whatever order?

• \displaystyle \frac{1}{8}81โ
• \displaystyle \frac{3}{8}83โ
• \displaystyle \frac{1}{4}41โ
• \displaystyle \frac{1}{2}21โ
• \displaystyle \frac{5}{8}85โ
• \displaystyle \frac{7}{8}87โ

Q2. Letโs play a game! You toss a (fair) coin. If it comes out heads, you win. Otherwise, the turn passes on to PLAYER 2, who tosses the same coin and wins if it comes out heads. If not, it is PLAYER 3โs turn. If she doesnโt get heads either, it is your turn again. The game goes on until somebody gets heads. What is the probability that you win?

• \displaystyle \frac{1}{2} + \frac{1}{2^4} + \frac{1}{2^7} + \cdots = \frac{4}{7}21โ+241โ+271โ+โฏ=74โ
• \displaystyle \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \cdots = 121โ+221โ+231โ+โฏ=1
• \displaystyle \frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + \cdots = \frac{1}{2}31โ+321โ+331โ+โฏ=21โ
• \displaystyle \frac{1}{2} + \frac{1}{2^3} + \frac{1}{2^5} + \cdots = \frac{2}{3}21โ+231โ+251โ+โฏ=32โ
• \displaystyle \frac{1}{3}31โ
• 00

Q3. A bus line runs every 30 minutes. If you arrive at a stop randomly, what is the probability that you will have to wait more than 10 minutes for the next bus?

Hint: this probability is a โvolumeโ fraction over some domain. What is the domain, and what is its dimension?

• \displaystyle \frac{1}{4}41โ
• \displaystyle \frac{1}{3}31โ
• 11
• \displaystyle \frac{2}{3}32โ
• \displaystyle \frac{3}{4}43โ
• \displaystyle \frac{1}{2}21โ

Q4. What is the probability that a randomly chosen point of a square of side length LL is more than a distance rr away from every corner? Suppose r \lt L/2r<L/2.

• \displaystyle L^2 โ \frac{\pi r^2}{4}L2โ4ฯr2โ
• \displaystyle \frac{\pi r^2}{L^2}L2ฯr2โ
• \displaystyle \pi\left(\frac{r}{L}\right)^2 โ 1ฯ(Lrโ)2โ1
• L^2 โ \pi r^2L2โฯr2
• \displaystyle 1 โ \pi\left(\frac{r}{L}\right)^21โฯ(Lrโ)2
• 1 โ \displaystyle\frac{\pi r^2}{L}1โLฯr2โ

Q5. In the lecture we found out that the probability that a randomly chosen point in a square lies within its inscribed circle (see the figure on the left) is

P = \frac{\text{area of the disc}}{\text{area of the square}} = \frac{\pi r^2}{(2r)^2} = \frac{\pi}{4},P=area of the squarearea of the discโ=(2r)2ฯr2โ=4ฯโ,

where rr is the radius of the circle. Notice that this probability is independent of rr !

Reasoning in the same way, compute the probability that a randomly chosen point in a disc lies within its inscribed square (see the figure on the right).

• \displaystyle \frac{2}{\pi}ฯ2โ
• \displaystyle \frac{\pi}{4}4ฯโ
• \displaystyle \frac{4}{\pi}ฯ4โ
• \displaystyle \frac{\pi}{2}2ฯโ
• \displaystyle \frac{\sqrt{2}}{\pi r}ฯr2โโ
• \displaystyle \frac{\pi r}{\sqrt{2}}2โฯrโ

#### Quiz 02: Core Homework: Probability Densities

Q1. Which of the following cannot be a probability density function on the domain given? Select all that apply.

• ฯ(n)={52โโ51โโif n evenif n oddโ on n = 0, 1, \ldots, 9n=0,1,โฆ,9.
• \displaystyle \rho(n) = \frac{1}{10}ฯ(n)=101โ on n = 0, 1, \ldots, 10n=0,1,โฆ,10.
• \displaystyle \rho(n) =

โงโฉโจ150if n evenif n odd

• ฯ(n)={51โ0โif n evenif n oddโ on n = 0, 1, \ldots, 9n=0,1,โฆ,9.

\displaystyle \rho(n) = \frac{1}{n}ฯ(n)=n1โ on n = 1, 2, \ldotsn=1,2,โฆ

• \displaystyle \rho(n) = \frac{1}{10}ฯ(n)=101โ on n = 0, 1, \ldots, 9n=0,1,โฆ,9.

\displaystyle \rho(n) =

{10if n=1otherwise

• ฯ(n)={10โif n=1otherwiseโ on n = 1, 2, \ldotsn=1,2,โฆ

Q2. Which of the following cannot be a probability density function on the domain given? Select all that apply.

• \displaystyle \rho(x) = \frac{1}{10}ฯ(x)=101โ on [0, 10][0,10]
• \displaystyle \rho(x) = \frac{2}{\pi} \frac{1}{1+x^2}ฯ(x)=ฯ2โ1+x21โ on [0, +\infty)[0,+โ).
• \displaystyle \rho(x) = \frac{1}{2\pi} + \sin xฯ(x)=2ฯ1โ+sinx on [0, 2\pi][0,2ฯ]
• \displaystyle \rho(x) = \frac{2}{\pi} \frac{1}{1+x^2}ฯ(x)=ฯ2โ1+x21โ on \mathbb{R} = (-\infty, +\infty)R=(โโ,+โ).
• \displaystyle \rho(x) = \frac{1}{x^2}ฯ(x)=x21โ on [1, +\infty)[1,+โ).
• \displaystyle \rho(x) = \frac{1}{10}ฯ(x)=101โ on [0, 9][0,9]

Q3. For which value of \lambdaฮป is \rho(x) = \lambda x^2 e^{-x}ฯ(x)=ฮปx2eโx a probability density function on [0, +\infty)[0,+โ) ?

• \rho(x)ฯ(x) is not a probability density function for any value of \lambdaฮป
• \displaystyle \lambda = \frac{e}{2}ฮป=2eโ
• \displaystyle \lambda = \frac{1}{e}ฮป=e1โ
• \lambda = 1ฮป=1
• \displaystyle \lambda = \frac{1}{2}ฮป=21โ
• \lambda = 2ฮป=2

Q4. The amount of time between failures of a printer follows an exponential probability distribution โthat is, right after being repaired, the probability that the printer will fail after a time at most TT is given by

\int_{t=0}^T \alpha e^{-\alpha t} \, dtโซt=0Tโฮฑeโฮฑtdt

for \alpha = 0.01\ln 2\,\,\, \mathrm{h}^{-1}ฮฑ=0.01ln2hโ1 (notice that \alphaฮฑ has units of inverse time, in this case, inverse hours). What is the probability that the printer does not fail for 200\, \mathrm{h}200h after the last repair?

• 1 โ e^{-2}1โeโ2
• \displaystyle \frac{1}{4}41โ
• \displaystyle \frac{1}{2}21โ
• \displaystyle \frac{3}{4}43โ
• e^{-1/2}eโ1/2
• e^{-2}eโ2

#### Quiz 03: Core Homework: Expectation and Variance

Q1. Find the expectation \mathbb{E}E and variance \mathbb{V}V of xx if its probability density function is \rho(x) = (n+1) x^nฯ(x)=(n+1)xn (nn a positive integer) on [0, 1][0,1].

• \displaystyle \mathbb{E} = \frac{n+1}{n+2}E=n+2n+1โ, \displaystyle \mathbb{V} = \frac{n+1}{n+3} โ \left( \frac{n+1}{n+2} \right)^2V=n+3n+1โโ(n+2n+1โ)2.
• \mathbb{E} = 1E=1, \displaystyle \mathbb{V} = \frac{n+1}{n+2}V=n+2n+1โ.
• \mathbb{E} = 1E=1, \displaystyle \mathbb{V} = \frac{n+1}{n+2} โ 1V=n+2n+1โโ1.
• \mathbb{E} = 1E=1, \displaystyle \mathbb{V} = \frac{n+2}{n+3}V=n+3n+2โ.
• \displaystyle \mathbb{E} = \frac{n+1}{n+2}E=n+2n+1โ, \displaystyle \mathbb{V} = \frac{n+2}{n+3}V=n+3n+2โ.
• \displaystyle \mathbb{E} = \frac{n+1}{n+2}E=n+2n+1โ, \displaystyle \mathbb{V} = \frac{n+2}{n+3} โ \left( \frac{n+1}{n+2} \right)^2V=n+3n+2โโ(n+2n+1โ)2.

Q2 .Find the expectation \mathbb{E}E and variance \mathbb{V}V of xx if its probability density function is \displaystyle \rho(x) = \frac{2}{\pi} \frac{1}{x^2 + 1}ฯ(x)=ฯ2โx2+11โ on [0, +\infty)[0,+โ).

Hint: notice that the integrals calculating the expectation and variance are improper because [0, +\infty)[0,+โ) is unbounded. The first thing you should always do when confronted with one of these is check whether it converges or not.

• \displaystyle \mathbb{E} = \frac{2}{\pi}E=ฯ2โ, but \mathbb{V}V diverges.
• \mathbb{E} = 1E=1, \mathbb{V} = 1V=1.
• \displaystyle \mathbb{E} = \frac{2}{\pi}E=ฯ2โ, \displaystyle \mathbb{V} = \frac{2}{\pi} โ \frac{4}{\pi^2}V=ฯ2โโฯ24โ.
• \displaystyle \mathbb{E} = \frac{2}{\pi}E=ฯ2โ, \displaystyle \mathbb{V} = \frac{4}{\pi^2}V=ฯ24โ.
• \mathbb{E} = 1E=1, but \mathbb{V}V diverges.
• Both \mathbb{E}E and \mathbb{V}V diverge.

Q3. Find the expectation \mathbb{E}E and variance \mathbb{V}V of nn if its probability density function is \displaystyle \rho(n) = \frac{1}{4}ฯ(n)=41โ on n = 1, 2, 3, 4n=1,2,3,4.

Hint: although we have not talked about expectation and variance for discrete probability distributions, you can do this! Think of the analogy with masses: expectation is center of mass and variance is moment of inertia. This problem hints at the fact that you can think of sums as discrete versions of integrals, opening the door to using Calculus in situations in which inputs are discrete but outputs are continuous. Much more about this in Chapter 5: Discretization!

• \mathbb{E} = 2E=2, \displaystyle \mathbb{V} = \frac{7}{2}V=27โ.
• \mathbb{E} = 2E=2, \displaystyle \mathbb{V} = \frac{5}{4}V=45โ.
• \displaystyle \mathbb{E} = \frac{5}{2}E=25โ, \displaystyle \mathbb{V} = \frac{7}{2}V=27โ.
• \displaystyle \mathbb{E} = \frac{5}{2}E=25โ, \displaystyle \mathbb{V} = \frac{5}{4}V=45โ.
• \displaystyle \mathbb{E} = \frac{5}{2}E=25โ, \displaystyle \mathbb{V} = \frac{15}{4}V=415โ.
• \mathbb{E} = 2E=2, \displaystyle \mathbb{V} = \frac{1}{2}V=21โ.

Q4. The median mm of a (one-dimensional) continuous probability distribution on [a,b][a,b] is defined to be the value of xx for which the probability of x \lt mx<m is equal to the probability of x \gt mx>m โthat is, 1/21/2. In the language of integrals, this is:

\int_a^m\rho(x)dx = \frac{1}{2} = \int_m^b\rho(x)dxโซamโฯ(x)dx=21โ=โซmbโฯ(x)dx

Find the value of the median for an exponential distribution with probability density function

\rho(x) = \alpha e^{-\alpha x} \qquad \text{on}\:\:[0, +\infty)ฯ(x)=ฮฑeโฮฑxon[0,+โ)

• m = \alpham=ฮฑ
• \displaystyle m = \alpha \ln 2m=ฮฑln2
• \displaystyle m = \frac{\alpha}{\ln 2}m=ln2ฮฑโ
• \displaystyle m = \frac{1}{\alpha \ln 2}m=ฮฑln21โ
• \displaystyle m = \frac{\ln 2}{\alpha}m=ฮฑln2โ
• \displaystyle m = \frac{1}{\alpha}m=ฮฑ1โ

Q5. There is a host of other numbers that one can associate to a probability distribution that generalize the median: e.g., the so-called quantiles. Let us consider an example โthe quartiles:

• the first (or lower) quartile is the unique value Q_1Q1โ for which the probability of x \lt Q_1x<Q1โ is 1/41/4;
• the second quartile (really the median) is the unique value Q_2Q2โ for which the probability of x \lt Q_2x<Q2โ is 2/42/4;
• the third (or upper) quartile is the unique value Q_3Q3โ for which the probability of x \lt Q_3x<Q3โ is 3/43/4;

You also have quintilesdeciles, the ubiquitous percentiles, etc.

Find the value of the first and third quartiles of the exponential distribution of the previous problem with \rho(x)=\alpha e^{-\alpha x}ฯ(x)=ฮฑeโฮฑx.

• \displaystyle Q_1 = \frac{1}{\alpha \ln 4}Q1โ=ฮฑln41โ, \displaystyle Q_3 = \frac{1}{\alpha \ln (4/3)}Q3โ=ฮฑln(4/3)1โ.
• \displaystyle Q_1 = \frac{\alpha}{\ln 4}Q1โ=ln4ฮฑโ, \displaystyle Q_3 = \frac{\alpha}{\ln (4/3)}Q3โ=ln(4/3)ฮฑโ.
• \displaystyle Q_1 = \frac{\ln (4/3)}{\alpha}Q1โ=ฮฑln(4/3)โ, \displaystyle Q_3 = \frac{\ln 4}{\alpha}Q3โ=ฮฑln4โ.
• \displaystyle Q_1 = \frac{\alpha}{\ln (4/3)}Q1โ=ln(4/3)ฮฑโ, \displaystyle Q_3 = \frac{\alpha}{\ln 4}Q3โ=ln4ฮฑโ.
• \displaystyle Q_1 = \frac{1}{\alpha \ln (4/3)}Q1โ=ฮฑln(4/3)1โ, \displaystyle Q_3 = \frac{1}{\alpha \ln 4}Q3โ=ฮฑln41โ.
• \displaystyle Q_1 = \frac{\ln 4}{\alpha}Q1โ=ฮฑln4โ, \displaystyle Q_3 = \frac{\ln (4/3)}{\alpha}Q3โ=ฮฑln(4/3)โ.

#### Quiz 04:Chapter 4: Applications โ Exam

Q1. Compute the expectation \mathbb{E}E of xx with the probability density function

\rho(x) = \frac{3}{2}\sqrt{x}ฯ(x)=23โxโ

on 0 \leq x \leq 10โคxโค1.

• \displaystyle \frac{5}{3}35โ
• \displaystyle \frac{2}{3}32โ
• \displaystyle \frac{1}{2}21โ
• \displaystyle \frac{2}{5}52โ
• \displaystyle \frac{4}{15}154โ
• \displaystyle \frac{3}{5}53โ

Q2. An aerosol spray releases spherical droplets whose radii are distributed randomly by a uniform distribution between 11 and 33 micrometers.

What is the average volume of such an aerosol droplet (in units of cubic micrometers)?

Hint: The volume of the average-radius droplet is not necessarily the average volumeโฆ

• \displaystyle\frac{80}{3}\pi380โฯ
• 9\pi9ฯ
• \displaystyle\frac{13}{3}\pi313โฯ
• \displaystyle\frac{26}{3}\pi326โฯ
• \displaystyle\frac{80}{9}\pi980โฯ
• \displaystyle\frac{40}{3}\pi340โฯ

Q3. Find the yy-coordinate of the center of mass of a thin sheet of metal of constant density of a shape bounded by the xx-axis and the parabola

y= 1 โ \frac{x^2}{25}y=1โ25x2โ

• \displaystyle \frac{2}{5}52โ
• \displaystyle \frac{8}{3}38โ
• 00
• \displaystyle \frac{4}{5}54โ
• \displaystyle \frac{4}{3}34โ
• \displaystyle \frac{8}{5}58โ

Q4. Consider a swimming pool of some shape (with vertical sides, so that horizontal cross-sections have the same shape). Assume that it is completely full of water, and that it takes WW units of work to pump out all the water from the pool (pumping out to the elevation at the top of the pool). How much work did it take to pump out the first half of the water from the pool?

• \displaystyle\frac{1}{3}W31โW
• \displaystyle\frac{2}{\sqrt{2}}W2โ2โW
• \displaystyle\frac{1}{8}W81โW
• There is not enough information to answer this question.
• \displaystyle\frac{1}{\sqrt{2}}W2โ1โW
• \displaystyle\frac{1}{4}W41โW

Q5. Compute the moment of inertia II of a solid cylinder of mass MM, radius RR, and height hh about the central axis (passing through the centers of the cross-sectional discs).

• I = \displaystyle \frac{1}{2}MR^2I=21โMR2
• I = \displaystyle \frac{2}{3}MR^2I=32โMR2
• I = \displaystyle \frac{1}{2}MR^2hI=21โMR2h
• I = \displaystyle \frac{2}{3}MRhI=32โMRh
• I = \displaystyle \frac{1}{3}MRhI=31โMRh
• I = \displaystyle \frac{1}{2}MRhI=21โMRh

Q6. Find the volume of the body obtained by rotating about the xx-axis the region between the cuspidal cubic x^2 = y^3x2=y3, the xx-axis and the lines x=0x=0 and x=1x=1. Hint: you do not need a picture to solve this problemโฆ

• \displaystyle \frac{3\pi}{5}53ฯโ
• \displaystyle \frac{\pi}{5}5ฯโ
• \displaystyle \frac{3\pi}{7}73ฯโ
• \displaystyle \frac{\pi}{8}8ฯโ
• \displaystyle \frac{9\pi}{7}79ฯโ
• \displaystyle \frac{\pi}{7}7ฯโ

Q7. What is the area in the plane enclosed by the graph of the function r(\theta) = \cos \theta + \sin \thetar(ฮธ)=cosฮธ+sinฮธ (defined using polar coordinates) for \thetaฮธ between 00 and 3\pi/43ฯ/4?

• \displaystyle \frac{1}{4}41โ
• \displaystyle \frac{1 + \sqrt{2}}{2}21+2โโ
• 1 + \sqrt{2}1+2โ
• \displaystyle \frac{3\pi}{4}+\frac{1}{2}43ฯโ+21โ
• \displaystyle \frac{3\pi}{8}+\frac{1}{4}83ฯโ+41โ
• \piฯ

Q8. Which one of the following integrals computes the surface area of the surface obtained by rotating a quarter-circle

x^2 + y^2 = 4, \qquad x, y \geq 0x2+y2=4,x,yโฅ0

Hint 1: slice into horizontal strips.

Hint 2: donโt integrate this! (though you could if you had toโฆ)

• \displaystyle \int_{x=0}^2 \sqrt{\frac{4}{4-x^2}} \, dxโซx=02โ4โx24โโdx
• \displaystyle \int_{x=0}^2 2\pi x \sqrt{\frac{4}{4-x^2}} \, dxโซx=02โ2ฯx4โx24โโdx
• \displaystyle \int_{x=-1}^2 2\pi x \sqrt{\frac{4}{4-x^2}} \, dxโซx=โ12โ2ฯx4โx24โโdx
• \displaystyle \int_{x=-1}^1 2\pi x \sqrt{\frac{4}{4-x^2}} \, dxโซx=โ11โ2ฯx4โx24โโdx
• \displaystyle \int_{x=0}^2 2\pi(x+1)\sqrt{\frac{4}{4-x^2}} \, dxโซx=02โ2ฯ(x+1)4โx24โโdx
• \displaystyle \int_{x=0}^2 2\pi(x+1)\sqrt{1+4x^2} \, dxโซx=02โ2ฯ(x+1)1+4x2โdx

Q9. Find the arc length of the curve \displaystyle y = \frac{x^2}{4} โ \frac{\ln x}{2}y=4x2โโ2lnxโ between x=1x=1 and x=ex=e.

Hint: if you compute the length element correctly, a miraculous simplification should occur, making the integral doable.

• \displaystyle \frac{e^2 + 1}{4}4e2+1โ
• \displaystyle \frac{2\pi e}{3}32ฯeโ
• \displaystyle \frac{e^2 โ 2}{4}4e2โ2โ
• \displaystyle \frac{e^2 + 2}{4}4e2+2โ
• \displaystyle \frac{e^2 โ 1}{4}4e2โ1โ
• \displaystyle \frac{e^2}{4}4e2โ

Q10. the present value PVPV of the following income stream I(t)I(t), assuming an continuously-compounding interest rate of 55 per cent (r=0.05r=0.05). The income stream is the following: for the first 1010 years, you get nothing: I(t)=0I(t)=0 for 0\leq t\leq 100โคtโค10. Then, you get income at a constant rate of ten-thousand (10,\!00010,000) dollars-per-year in perpetuity (that is, you get money at that rate for all future time).

• PV = \displaystyle \frac{200,\!000}{e}PV=e200,000โ
• PV = \displaystyle \frac{500}{\sqrt{e}}PV=eโ500โ
• PV = \displaystyle 100,\!000 e^2PV=100,000e2
• PV = 200,\!000PV=200,000
• PV = \displaystyle 5,\!000 ePV=5,000e
• PV = \displaystyle \frac{200,\!000}{\sqrt{e}}PV=eโ200,000โ

Calculus is one of the greatest things that people have thought of. It helps us understand everything from the orbits of planets to the best size for a city to how often a heart beats. This quick course covers the main ideas of Calculus with one variable, with a focus on understanding the ideas and how to use them. This course is perfect for students who are just starting out in engineering, the physical sciences, or the social sciences.

The course is different because:

1) Taylor series and approximations are introduced and used from the start;

2) a new way of combining discrete and continuous forms of calculus is used;

3) the emphasis is on the ideas rather than the calculations; and

4) the course is taught in a clear, dynamic, and unified way.

In this fourth part, part four of five, we talk about computing areas and volumes, other geometric applications, physical applications, averages and mass, and probability.

### 825 thoughts on “Calculus: Single Variable Part 4 โ Applications Coursera Quiz Answers 2022 | All Weeks Assessment Answers [๐ฏCorrect Answer]”

1. You have brought up a very superb points, thankyou for the post.

2. Very interesting information!Perfect just what I was searching for! “The only gift is a portion of thyself.” by Ralph Waldo Emerson.

3. Thank you for every other excellent post. The place else may just anybody get that kind of info in such an ideal approach of writing? I’ve a presentation subsequent week, and I am on the look for such info.

4. Thank you a bunch for sharing this with all of us you actually understand what you’re speaking approximately! Bookmarked. Kindly also consult with my web site =). We may have a link exchange arrangement between us!

5. I don’t usually comment but I gotta tell thankyou for the post on this perfect one : D.

6. An interesting dialogue is value comment. I believe that you need to write more on this topic, it may not be a taboo topic but usually persons are not sufficient to talk on such topics. To the next. Cheers

7. Thanks , I have recently been looking for info about this topic for ages and yours is the best I’ve discovered till now. But, what about the conclusion? Are you sure about the source?

8. I like this site very much, Its a very nice place to read and receive information.

9. But wanna state that this is very useful, Thanks for taking your time to write this.

10. With havin so much content do you ever run into any problems of plagorism or copyright violation? My site has a lot of completely unique content I’ve either written myself or outsourced but it looks like a lot of it is popping it up all over the internet without my agreement. Do you know any methods to help reduce content from being ripped off? I’d really appreciate it.

11. Simply wish to say your article is as surprising. The clearness in your post is just spectacular and i can assume you’re an expert on this subject. Fine with your permission allow me to grab your feed to keep up to date with forthcoming post. Thanks a million and please continue the enjoyable work.

12. great post.Never knew this, regards for letting me know.

13. I do agree with all the ideas you’ve presented in your post. They are really convincing and will certainly work. Still, the posts are too short for newbies. Could you please extend them a bit from next time? Thanks for the post.

14. Hi there! Do you know if they make any plugins to protect against hackers? I’m kinda paranoid about losing everything I’ve worked hard on. Any recommendations?

15. so much wonderful information on here, : D.

16. I do love the manner in which you have framed this particular problem plus it does indeed present me some fodder for consideration. Nonetheless, through just what I have personally seen, I just wish as other opinions pack on that folks stay on issue and not start upon a soap box associated with the news du jour. All the same, thank you for this superb piece and though I do not go along with the idea in totality, I regard your perspective.

17. A number of players over the years have told me they beat the slots by looking for larger than usual progressive jackpots. Itโs a method that works better on video poker where the house edge on the base game is smaller than on the slots. Slot payback percentages are low enough that even what looks like an oversized jackpot may not be enough to overcome the full house edge. Still, if you always wait to play a game until its jackpot is a certain size, you will be playing a game with a lower house edge than if you played for lesser amounts. Super 8 Ways Ultimate is an online casino game that definitely doesnโt feel like one โ that is, youโd be more used to seeing something like this on a physical slot machine a decade or two ago, not on the present-day internet. That said, we donโt think itโs 100% safe to say itโs not worth your attention โ we know there are a lot of fans of old-school casino games out there.
http://www.rpec.co.kr/bbs/board.php?bo_table=free&wr_id=89410
This site has limited support for your browser. We recommend switching to Edge, Chrome, Safari, or Firefox. Since 2020, we have been awarding the best poker operators and innovators in the poker business with our PokerListings Operator Awards. Every year the operators are nominated and awarded in… Categories \$9,800.00 Jimmy, we received our dining poker table and were amazed at the quality, the detail and the excellent craftsmanship. It was certainly beyond our expectations. And, the assembly was easy with two persons. Everyone that sees our dining table is shocked to see the dining surface removed to display the beauty of the Texas Holdem layout underneath. You did a magnificent job building our table as we had discussed. We absolutely love it and so do our friends. We highly recommend you to build the perfect combination table. Thank you so very much.

18. I like this web blog very much, Its a really nice spot to read and get information. “If at first you don’t succeed, you’re running about average.” by M. H. Alderson.

19. This really answered my downside, thanks!

20. I happen to be commenting to let you know of the extraordinary experience our child went through visiting your web site. She mastered some details, with the inclusion of how it is like to have a marvelous helping mood to make most people smoothly learn specified problematic issues. You undoubtedly exceeded our desires. Thanks for distributing such necessary, trusted, educational and even easy tips on this topic to Mary.

21. This blog is definitely rather handy since Iโm at the moment creating an internet floral website โ although I am only starting out therefore itโs really fairly small, nothing like this site. Can link to a few of the posts here as they are quite. Thanks much. Zoey Olsen

22. Wow! This blog looks just like my old one! It’s on a completely different subject but it has pretty much the same page layout and design. Outstanding choice of colors!

23. I’ve been surfing on-line greater than 3 hours lately, but I never discovered any attention-grabbing article like yours. It is beautiful value enough for me. Personally, if all webmasters and bloggers made excellent content as you probably did, the net will likely be a lot more useful than ever before. “Baseball is 90 percent mental. The other half is physical.” by Lawrence Peter Berra.

24. I like this site so much, saved to my bookmarks. “American soldiers must be turned into lambs and eating them is tolerated.” by Muammar Qaddafi.

25. Wonderful work! This is the type of information that are meant to be shared around the web. Disgrace on Google for now not positioning this put up upper! Come on over and visit my web site . Thank you =)

26. This web site is my inhalation, real excellent layout and perfect subject matter.

27. As I website possessor I believe the articles here is real superb, appreciate it for your efforts.

28. Itโs hard to find knowledgeable people on this topic, but you sound like you know what youโre talking about! Thanks

29. Great line up. We will be linking to this great article on our site. Keep up the good writing.

30. Keep up the wonderful piece of work, I read few posts on this website and I think that your web site is really interesting and has got lots of wonderful info .

31. But a smiling visitor here to share the love (:, btw outstanding design.

32. I have been browsing on-line more than three hours today, but I by no means discovered any fascinating article like yours. It?ยฆs pretty price enough for me. In my opinion, if all web owners and bloggers made just right content material as you did, the net shall be much more useful than ever before.

33. To announce actual scoop, follow these tips:

Look in behalf of credible sources: http://mylifestyle.us/wp-content/pgs/how-to-remove-taboola-news-from-android-phone.html. It’s high-ranking to secure that the newscast origin you are reading is reliable and unbiased. Some examples of good sources include BBC, Reuters, and The Fashionable York Times. Interpret multiple sources to get a well-rounded aspect of a isolated info event. This can help you carp a more ended paint and avoid bias. Be aware of the angle the article is coming from, as constant reputable telecast sources can compel ought to bias. Fact-check the dirt with another fountain-head if a scandal article seems too staggering or unbelievable. Forever be persuaded you are reading a current article, as exposโ can change-over quickly.

Nearby following these tips, you can befit a more informed news reader and best be aware the beget here you.

34. Positively! Find news portals in the UK can be crushing, but there are scads resources at to cure you think the best one for you. As I mentioned formerly, conducting an online search an eye to https://kitjohnson.co.uk/pag/learn-how-to-outsmart-fake-news.html “UK news websites” or “British news portals” is a enormous starting point. Not but determination this grant you a encompassing slate of hearsay websites, but it will also lend you with a improved understanding of the current communication landscape in the UK.
On one occasion you be enduring a file of embryonic account portals, it’s powerful to gauge each sole to influence which richest suits your preferences. As an exempli gratia, BBC News is known for its intention reporting of intelligence stories, while The Guardian is known representing its in-depth criticism of political and social issues. The Self-governing is known for its investigative journalism, while The Times is known in the interest of its vocation and wealth coverage. By way of understanding these differences, you can select the information portal that caters to your interests and provides you with the news you call for to read.
Additionally, it’s usefulness considering neighbourhood despatch portals for proper to regions within the UK. These portals lay down coverage of events and good copy stories that are fitting to the area, which can be specially cooperative if you’re looking to keep up with events in your local community. In place of exemplar, local communiquโ portals in London contain the Evening Paradigm and the Londonist, while Manchester Evening Talk and Liverpool Reflection are in demand in the North West.
Comprehensive, there are diverse tidings portals readily obtainable in the UK, and it’s high-ranking to do your inspection to unearth the everybody that suits your needs. At near evaluating the contrasting news broadcast portals based on their coverage, luxury, and article standpoint, you can select the individual that provides you with the most fitting and captivating info stories. Good destiny with your search, and I anticipation this tidings helps you reveal the correct news portal inasmuch as you!

35. Wow, fantastic weblog format! How long have you been running a blog for? you made running a blog glance easy. The overall look of your site is wonderful, as well as the content material!

36. Your article gave me a lot of inspiration, I hope you can explain your point of view in more detail, because I have some doubts, thank you. 20bet

37. I feel that is among the so much important info for me.
And i am satisfied studying your article. But should commentary on few basic things, The site style is ideal, the articles is
actually great : D. Excellent process, cheers

38. I am glad to be one of many visitors on this outstanding internet site (:, thankyou for putting up.

39. I like what you guys are up also. Such smart work and reporting! Carry on the superb works guys I’ve incorporated you guys to my blogroll. I think it will improve the value of my site :).

40. Superb, what a blog it is! This web site provides helpful data to us, keep it up.

41. order amoxicillin 500mg sale oral trimox order clarithromycin 500mg online cheap

42. buy letrozole 2.5mg generic abilify order buy aripiprazole 20mg without prescription

43. ortexi is a 360ยฐ hearing support designed for men and women who have experienced hearing loss at some point in their lives.

44. FitSpresso is a special supplement that makes it easier for you to lose weight. It has natural ingredients that help your body burn fat better.

45. This really answered my drawback, thanks!

46. Aizen Power is a cutting-edge male enhancement formula that improves erections and performance. The supplement contains only natural ingredients derived from organic plants.

47. Arteris Plus is a revolutionary supplement designed to support individuals struggling with high blood pressure, also known as the silent killer.

48. Boostaro increases blood flow to the reproductive organs, leading to stronger and more vibrant erections. It provides a powerful boost that can make you feel like you’ve unlocked the secret to firm erections

49. ErecPrime is a 100% natural supplement which is designed specifically

50. Neotonics is a dietary supplement that offers help in retaining glowing skin and maintaining gut health for its users. It is made of the most natural elements that mother nature can offer and also includes 500 million units of beneficial microbiome.

51. ProDentim is a nutritional dental health supplement that is formulated to reverse serious dental issues and to help maintain good dental health.

52. Hello there I am so glad I found your webpage, I really found you by mistake, while I was researching on Askjeeve for something else, Nonetheless I am here now and would just like to say many thanks for a remarkable post and a all round interesting blog (I also love the theme/design), I donโt have time to go through it all at the minute but I have saved it and also added your RSS feeds, so when I have time I will be back to read much more, Please do keep up the excellent job.

53. Do you have a spam problem on this site; I also am a blogger, and I was wondering your situation; many of us have developed some nice procedures and we are looking to trade methods with other folks, please shoot me an e-mail if interested.

54. SonoViveโข is a completely natural hearing support formula made with powerful ingredients that help heal tinnitus problems and restore your hearing

55. BioFitโข is a Nutritional Supplement That Uses Probiotics To Help You Lose Weight

56. GlucoCare is a natural and safe supplement for blood sugar support and weight management. It fixes your metabolism and detoxifies your body.

57. Nervogen Pro, A Cutting-Edge Supplement Dedicated To Enhancing Nerve Health And Providing Natural Relief From Discomfort. Our Mission Is To Empower You To Lead A Life Free From The Limitations Of Nerve-Related Challenges. With A Focus On Premium Ingredients And Scientific Expertise.

58. Neurodrine is a fantastic dietary supplement that protects your mind and improves memory performance. It can help you improve your focus and concentration.

59. InchaGrow is an advanced male enhancement supplement. The Formula is Easy to Take Each Day, and it Only Uses. Natural Ingredients to Get the Desired Effect

60. Amiclear is a dietary supplement designed to support healthy blood sugar levels and assist with glucose metabolism. It contains eight proprietary blends of ingredients that have been clinically proven to be effective.

61. Metabo Flex is a nutritional formula that enhances metabolic flexibility by awakening the calorie-burning switch in the body. The supplement is designed to target the underlying causes of stubborn weight gain utilizing a special โmiracle plantโ from Cambodia that can melt fat 24/7.

62. SynoGut is an all-natural dietary supplement that is designed to support the health of your digestive system, keeping you energized and active.

63. GlucoFlush Supplement is an all-new blood sugar-lowering formula. It is a dietary supplement based on the Mayan cleansing routine that consists of natural ingredients and nutrients.

64. Herpagreens is a dietary supplement formulated to combat symptoms of herpes by providing the body with high levels of super antioxidants, vitamins

65. Introducing FlowForce Max, a solution designed with a single purpose: to provide men with an affordable and safe way to address BPH and other prostate concerns. Unlike many costly supplements or those with risky stimulants, we’ve crafted FlowForce Max with your well-being in mind. Don’t compromise your health or budget โ choose FlowForce Max for effective prostate support today!

66. TerraCalm is an antifungal mineral clay that may support the health of your toenails. It is for those who struggle with brittle, weak, and discoloured nails. It has a unique blend of natural ingredients that may work to nourish and strengthen your toenails.

67. I’m truly enjoying the design and layout of your website. It’s a very easy on the eyes which makes it much more enjoyable for me to come here and visit more often. Did you hire out a designer to create your theme? Outstanding work!

68. Cortexi is an effective hearing health support formula that has gained positive user feedback for its ability to improve hearing ability and memory. This supplement contains natural ingredients and has undergone evaluation to ensure its efficacy and safety. Manufactured in an FDA-registered and GMP-certified facility, Cortexi promotes healthy hearing, enhances mental acuity, and sharpens memory.

69. Congratulations on your incredible gift for writing! Your article is an engaging and enlightening read. Wishing you a New Year full of achievements and happiness!

70. Magnificent article! If you require a writer, I’m here and passionate about writing

71. The article was captivating. To further engage your readers, consider adding more visuals. My website might have some resources to help.

72. The article was very engaging. Consider incorporating more visuals to make it stand out. My website can offer some suggestions.

75. Sight Care is a natural supplement designed to improve eyesight and reduce dark blindness. With its potent blend of ingredients. https://sightcarebuynow.us/

76. I enjoyed reading your article! The information is presented effectively. Adding more visuals in your next articles could make the content more engaging for readers like me.

77. Well done! ๐ Your article is informative and insightful. Including more images in your future articles could make the content more visually appealing. ๐จ

78. Artikel ini luar biasa! Cara menerangkan hal-hal sungguh memikat dan sangat mudah untuk dipahami. Sudah terlihat bahwa telah banyak usaha dan penyelidikan yang dilakukan, yang sungguh layak diapresiasi. Penulis berhasil membuat tema ini tidak hanya menarik tetapi juga menyenangkan untuk dibaca. Saya dengan antusias menantikan untuk eksplorasi konten seperti ini di masa depan. Terima kasih atas berbagi, Anda melakukan tugas yang luar biasa!

79. EndoPump is a dietary supplement for men’s health. This supplement is said to improve the strength and stamina required by your body to perform various physical tasks. Because the supplement addresses issues associated with aging, it also provides support for a variety of other age-related issues that may affect the body. https://endopumpbuynow.us/

80. BioFit is an all-natural supplement that is known to enhance and balance good bacteria in the gut area. To lose weight, you need to have a balanced hormones and body processes. Many times, people struggle with weight loss because their gut health has issues. https://biofitbuynow.us/

81. AquaPeace is an all-natural nutritional formula that uses a proprietary and potent blend of ingredients and nutrients to improve overall ear and hearing health and alleviate the symptoms of tinnitus. https://aquapeacebuynow.us/

82. Amiclear is a dietary supplement designed to support healthy blood sugar levels and assist with glucose metabolism. It contains eight proprietary blends of ingredients that have been clinically proven to be effective. https://amiclearbuynow.us/

83. Herpagreens is a dietary supplement formulated to combat symptoms of herpes by providing the body with high levels of super antioxidants, vitamins

84. Nervogen Pro is an effective dietary supplement designed to help patients with neuropathic pain. When you combine exotic herbs, spices, and other organic substances, your immune system will be strengthened. https://nervogenprobuynow.us/

85. Cortexi is a completely natural product that promotes healthy hearing, improves memory, and sharpens mental clarity. Cortexi hearing support formula is a combination of high-quality natural components that work together to offer you with a variety of health advantages, particularly for persons in their middle and late years. https://cortexibuynow.us/

86. Endopeak is a natural energy-boosting formula designed to improve men’s stamina, energy levels, and overall health. The supplement is made up of eight high-quality ingredients that address the underlying cause of declining energy and vitality. https://endopeakbuynow.us/

87. LeanBliss is a unique weight loss formula that promotes optimal weight and balanced blood sugar levels while curbing your appetite, detoxifying, and boosting your metabolism. https://leanblissbuynow.us/

88. Unlock the incredible potential of Puravive! Supercharge your metabolism and incinerate calories like never before with our unique fusion of 8 exotic components. Bid farewell to those stubborn pounds and welcome a reinvigorated metabolism and boundless vitality. Grab your bottle today and seize this golden opportunity! https://puravivebuynow.us/

89. GlucoTrust is a revolutionary blood sugar support solution that eliminates the underlying causes of type 2 diabetes and associated health risks. https://glucotrustbuynow.us/

90. Kerassentials are natural skin care products with ingredients such as vitamins and plants that help support good health and prevent the appearance of aging skin. Theyโre also 100% natural and safe to use. The manufacturer states that the product has no negative side effects and is safe to take on a daily basis. Kerassentials is a convenient, easy-to-use formula. https://kerassentialsbuynow.us/

91. Neotonics is an essential probiotic supplement that works to support the microbiome in the gut and also works as an anti-aging formula. The formula targets the cause of the aging of the skin. https://neotonicsbuynow.us/

92. Java Burn is a proprietary blend of metabolism-boosting ingredients that work together to promote weight loss in your body. https://javaburnbuynow.us/

93. LeanBiome is designed to support healthy weight loss. Formulated through the latest Ivy League research and backed by real-world results, it’s your partner on the path to a healthier you. https://leanbiomebuynow.us/

94. PowerBite is an innovative dental candy that promotes healthy teeth and gums. It’s a powerful formula that supports a strong and vibrant smile. https://powerbitebuynow.us/

95. Serolean, a revolutionary weight loss supplement, zeroes in on serotoninโthe key neurotransmitter governing mood, appetite, and fat storage. https://seroleanbuynow.us/

96. Neurozoom crafted in the United States, is a cognitive support formula designed to enhance memory retention and promote overall cognitive well-being. https://neurozoombuynow.us/

97. Quietum Plus supplement promotes healthy ears, enables clearer hearing, and combats tinnitus by utilizing only the purest natural ingredients. Supplements are widely used for various reasons, including boosting energy, lowering blood pressure, and boosting metabolism. https://quietumplusbuynow.us/

98. Researchers consider obesity a world crisis affecting over half a billion people worldwide. Vid Labs provides an effective solution that helps combat obesity and overweight without exercise or dieting. https://leanotoxbuynow.us/

99. Keratone addresses the real root cause of your toenail fungus in an extremely safe and natural way and nourishes your nails and skin so you can stay protected against infectious related diseases. https://keratonebuynow.us/

100. Sugar Defender is the #1 rated blood sugar formula with an advanced blend of 24 proven ingredients that support healthy glucose levels and natural weight loss. https://sugardefenderbuynow.us/

101. WONDERFUL Post.thanks for share..more wait .. โฆ

102. RVVR is website dedicated to advancing physical and mental health through scientific research and proven interventions. Learn about our evidence-based health promotion programs. https://rvvr.us/

103. Latest Denver news, top Colorado news and local breaking news from Denver News, including sports, weather, traffic, business, politics, photos and video. https://denver-news.us/

104. Island Post is the website for a chain of six weekly newspapers that serve the North Shore of Nassau County, Long Island published by Alb Media. The newspapers are comprised of the Great Neck News, Manhasset Times, Roslyn Times, Port Washington Times, New Hyde Park Herald Courier and the Williston Times. Their coverage includes village governments, the towns of Hempstead and North Hempstead, schools, business, entertainment and lifestyle. https://islandpost.us/

105. Looking for quick and easy dinner ideas? Browse 100

106. Healthcare Blog provides news, trends, jobs and resources for health industry professionals. We cover topics like healthcare IT, hospital administration, polcy

107. Stri is the leading entrepreneurs and innovation magazine devoted to shed light on the booming stri ecosystem worldwide. https://stri.us/

108. Valley News covers local news from Pomona to Ontario including, California news, sports, things to do, and business in the Inland Empire. https://valleynews.us/

109. Money Analysis is the destination for balancing life and budget – from money management tips, to cost-cutting deals, tax advice, and much more. https://moneyanalysis.us/

110. The destination for entertainment and women’s lifestyle – from royals news, fashion advice, and beauty tips, to celebrity interviews, and more. https://womenlifestyle.us/

111. The Boston Post is the leading source of breaking news, local news, sports, politics, entertainment, opinion and weather in Boston, Massachusetts. https://bostonpost.us/

112. Pilot News: Your source for Virginia breaking news, sports, business, entertainment, weather and traffic https://pilotnews.us/

113. Get Lehigh Valley news, Allentown news, Bethlehem news, Easton news, Quakertown news, Poconos news and Pennsylvania news from Morning Post. https://morningpost.us/

114. The latest food news: celebrity chefs, grocery chains, and fast food plus reviews, rankings, recipes, interviews, and more. https://todaymeal.us/

115. Foodie Blog is the destination for living a delicious life – from kitchen tips to culinary history, celebrity chefs, restaurant recommendations, and much more. https://foodieblog.us/

116. Guun specializes in informative deep dives – from history and crime to science and everything strange. https://guun.us/

117. indiaherald.us provides latest news from India , India News and around the world. Get breaking news alerts from India and follow todayโs live news updates in field of politics, business, sports, defence, entertainment and more. https://indiaherald.us

118. I have been browsing on-line more than 3 hours today, but I never
found any interesting article like yours.
It’s lovely value sufficient for me. Personally, if all
website owners and bloggers made good content material as you probably did, the
internet can be much more useful than ever before.

119. socialmediatric.com
Fang Jifan์ ๋ฏธ์๋ฅผ ์ง์ผ๋ฉฐ ๋งํ์ต๋๋ค. “์ ์ ํจ๊ป ์ ํ ์ฃผ๋ณ์ ์ฐ์ฑํ์ธ์.”

120. baseballoutsider.com
๊ทธ๋์ ์์นจ ์ผ์ฐ ์ผ์ด๋ ์ ์ท์ ์๊ณ  ๊ถ๊ถ๋ก ๋ค์ด๊ฐ์ต๋๋ค.

121. saungsantoso.com
๋ง์ ์ฅ์ธ๋ค์ด ์ฆ๊ธฐ ๊ธฐ๊ด์ฐจ๋ฅผ ๋ฐค์ ์ ๋นํ๊ณ  ์์ต๋๋ค.๋ง์ ํ๋ฆฐ๋ค์ด ์ฆ๊ฒ๊ฒ ๋ชจ์ฌ ํ ๋ก ์ ์์ํ์ต๋๋ค.

122. amruthaborewells.com
“๋ญ?” Fang Jifan์ Hongzhi ํฉ์ ๋ฅผ ๋ฐ๋ผ๋ณด๋ฉฐ ๋นํฉํ์ต๋๋ค.

123. Read the latest news on Crime, Politics, Schools, Sports, Weather, Business, Arts, Jobs, Obits, and Things to do in Kent Washington. https://kentnews.us/

124. Get Lehigh Valley news, Allentown news, Bethlehem news, Easton news, Quakertown news, Poconos news and Pennsylvania news from Morning Post. https://morningpost.us

125. tsrrub.com
ํ์ง๋ง… ์ด ํ๋ณต์ ๊ทธ๊ฒ์์ ์ค๋ ๊ฑด๋ฐ ์ ์ดํด๋ฅผ ๋ชป ํ ๊น์?

126. Metal scraps recycling Aluminium scrap reselling Scrap aluminium processing plant
Scrap metal marketing strategies, Aluminum cable shredding and size reduction, Metal waste brokerage

Metal industry trend analysis, Recycle aluminum cable, Metal utilization

128. Metal reclamation and recovery yard Recycling aluminium material handling Aluminium recycling chain management
Metal scrap dismantling, Aluminum cable stripping, Scrap metal reclaiming solutions

129. Scrap metal reclamation processing Scrap aluminium price trends Aluminium scrap sorting technologies
Metal scrap brokerage, Aluminum cable scrap recycling industry, Industrial scrap metal pricing

130. restaurant-lenvol.net
Hongzhi ํฉ์ ๋ ์ ์ ์นจ๋ฌตํ๋ค๊ฐ ์ค์ค๋ก ๋ ๋ฌ์ต๋๋ค.

131. baseballoutsider.com
๋ฐ๋ผ์ Fang Jifan์ด Mengxue ์ค๋ฆฝ์ ์ ์ํ์ ๋ ๋ง์ ์ฌ๋๋ค์ด ์จ์ ์ฌ๊ณ  ์ ํน์ ๋ฐ์์ต๋๋ค.

132. modernkarachi.com
Wang Shouren๊ณผ Tang Yin์ ์ผ์ ํ ์๋ก๋ฅผ ๋ฐ๋ผ๋ณด๋ค๊ฐ… ์ฌํธํก์ ํ์ต๋๋ค.

133. agonaga.com
๋น์ด๋จน์ ํ์ ์ด๋ฒ์๋ ๋งค์ฐ ์์ ์ ์ผ๋ก ๊ฒฝ๊ธฐ๋ฅผ ํผ์ณค์ต๋๋ค. ๊ณต์ ์ก์์ด๋ ๋น ๋ฅด๊ฒ ๊ณต๊ฒฉํ์ง ์์์ต๋๋ค.

134. smcasino7.com
์ค์จํฐ๋ฅผ ๋จ๊ฐ์งํ ๊ฒ์ ์ฒ์์ด์๊ณ , ์ด๊ฒ์ ๋จ๋๋ฐ ๋ณด๋ฆ์ด ๊ฑธ๋ ธ์ต๋๋ค.

135. sm-casino1.com
์ด๋ฆ์ด ๊ธด ์ด๊ฐ ์ฃผํ์กฐ๊ฐ ์ง์  ์น์งํ๋ค.

136. Bazopril is a blood pressure supplement featuring a blend of natural ingredients to support heart health