   Chapter 2.8, Problem 49E

Chapter
Section
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 m from the center of the track. How fast is the distance between the friends changing when the distance between them is 200 m?

To determine

To find:

How fast is the distance between the friends changing?

Explanation

1. Formula:

i. Arc length S=rθ

ii. Law of cosine:

c2=a2+b2-2abcosθ

2. Given:

Radius of circular track = 100 m Let S denote the distance covered by runner then

dSdt=7 m/s

The figure is as shown below.

3. Calculations:

Using arclength formula, S =rθ=100θ

Differentiate it.dsdt=100dθdt

Substituting dsdt=7=100dθdt=>dθdt=7100

Using cosine rule L2= a2+b2-abcosθ

L2= 1002+2002-2(100)(200)cosθ

Differentiate it with respect to t

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