Chapter 2.7, Problem 14E

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8th Edition

James Stewart

ISBN: 9781285740621

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

ii. Power rule of derivative:

**3) Given**:

And

**4) Calculation:**

Area of circle is

After t second radius of circle is

Substitute r = 60t in above formula,

Thus,

To find the rate after t seconds, take derivative of

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

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