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Math
CalculusSingle Variable Calculus: Early Transcendentals, Volume I8th Edition

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Chapter 3.9, Problem 49E
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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

Solutions

Chapter
Section
Problem 1E
Problem 2E
Problem 3E
Problem 4E
Problem 5E
Problem 6E
Problem 7E
Problem 8E
Problem 9E
Problem 10E
Problem 11E
Problem 12E
Problem 13E
Problem 14E
Problem 15E
Problem 16E
Problem 17E
Problem 18E
Problem 19E
Problem 20E
Problem 21E
Problem 22E
Problem 23E
Problem 24E
Problem 25E
Problem 26E
Problem 27E
Problem 28E
Problem 29E
Problem 30E
Problem 31E
Problem 32E
Problem 33E
Problem 34E
Problem 35E
Problem 36E
Problem 37E
Problem 38E
Problem 39E
Problem 40E
Problem 41E
Problem 42E
Problem 43E
Problem 44E
Problem 45E
Problem 46E
Problem 47E
Problem 48E
Problem 49E
Problem 50E
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

A runner sprints around a circular track of radius 100 m at a constant speed of 7 m/s. The runner's friend is standing at a distance 200 on from the center of the track. How fast is the distance between the friends changing when the distance between them is 200m?

To determine

To find: The rate of change of the distance between runner and his friend standing at

200 m from the center of the track.

Explanation

Given:

A runner sprints around a circular track of radius 100 m at a constant speed 7 m/s. And the runner`s is standing at a 200 m from the centre of the track.

Formula used:

(1) Chain rule: dydx=dydu⋅dudx

Calculation:

Let us assume that O be the center of the circular track of radius 100 m and P be the position of the runner on the circular track and A be the position of his friend from the 200 m from the center of the circular track.

Let θ be the angle at the center to the runner from the horizontal and l be the distance between the runner and his friend as shown in the Figure 1 given below.

Obtain dldt when l=200 m

Here, D is the arc length of the circular track, then

D=r×θ

Substitutes r =100.

D=100θθ=D100

Differentiate θ with respect to the time t.

dθdt=ddt(D100)=1100(dDdt)

Now from the cosine rule in the ΔOPA.

l2=1002+2002−(2×100×200cosθ)

Differentiate l2=1002+2002−(2×100×200cosθ) with respect to the time t.

ddt[l2]=ddt[1002+2002−(2×100 00cosθ)]2ldldt=−40,000(−sinθ)dθdt                                   [∵dydx=dydu⋅dudx]dldt=20,000l

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