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

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### 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=dydududx

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(2×10000cosθ)]2ldldt=40,000(sinθ)dθdt                                   [dydx=dydududx]dldt=20,000l

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Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 