   Chapter 2.7, Problem 14E

Chapter
Section
Textbook Problem

# A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s. Find the rate at which the area within the circle is increasing after (a) 1 s,(b) 3 s, and (c) 5 s. What can you conclude?

To determine

To find: The rate at which the area within the circle is increasing after (a) 1 s, (b) 3 s, and (c) 5 s then make conclusion.

Explanation

Conclusion:

We can conclude that, the rate of increase of area is linear with respect to time. Explanation:

1) Concept:

Use concept of derivative to find the rate after t seconds.

2) Formula:

i. Area of circle A= πr2

ii. Power rule of derivative:

ddxxn=nxn-1

3) Given:

drdt=60cms

And t=1s, 3s  and 5s

4) Calculation:

Area of circle is A= πr2

After t second radius of circle is  60t.

Substitute r = 60t in above formula,

Thus, A= π3600t2

To find the rate after t seconds, take derivative of A=π3600t2 with respect to t

By using power rule of derivative,

(a)To find the rate at which the area within the circle is increasing after 1 second,

Substitute t = 1 in above derivative.

Therefore, the area is increasing at rate 7200π cm2/s after 1 second

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