Question

Can someone show me how to do this step by step? 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 1s, 3s, and 5s. What can you conclude?

Answer 1

The area of a circle is \(A(r)=\pi r^2\), the rate at which the area is increasing is as follows:
\[1)\text{ after 1s }: \Delta_1={A(1)-A(0) \over 1-0}={\pi(60)^2-0 \over 1}=3600\pi.\]
\[2) \text{ after 3 s}: \Delta_2={A(3)-A(0) \over 3-0}={\pi(60(3))^2 \over 3}=3\pi(3600).\]\[3) \text{ after 5 s}: \Delta_3={A(5)-A(0) \over 5}={\pi(60(5))^2 \over 5}=5\pi(3600).\]

Answer 2

We can conclude that the area of the circle is increasing at a rate of \(3600\pi t \text{ cm}^2/s\), where t is time in seconds.

Answer 3

I made a slight mistake above in notations, but it doesn't change anything in the answers.

Answer 4

Thank you very much! =]

Answer 5

You're welcome.