Question

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 each of the following.

after 2s : cm2/s
after 5s : cm2/s
after 6s : cm2/s

Answer 1

9514 1404 393

Answer:

• 2s: 45,239 cm²/s
• 5s: 113,097 cm²/s
• 6s: 135,717 cm²/s

Explanation:

The radius is a function of time:

r(t) = 60t . . . . . radius in cm; time in s

Then the area of the circle is ...

A = πr² = π(60t)² = 3600πt²

The rate of change of area is the derivative of this:

A' = 2·3600πt = 7200πt

The rates of change of interest are ...

after 2s: 45,239 cm²/s

after 5s: 113,097 cm²/s

after 6s: 135,717 cm²/s