Math CentralQuandaries & Queries


Question from AnneMarie, a student:

A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 25 cm/s. Find the rate at which the area within the circle is increasing after 4s.

Hi Annemarie,

The area is a function of time and so is the radius so let the radius of the circular ripple be $r(t) \;$ cm and the area be $A(t) \; cm^2$ where $t$ is time measured in seconds. You know that

\[A(t) = \pi r^2.\]

Differentiate both sides with respect to $t$ and you will find $A'(t)$ expressed in terms of $t$ and $r'(t).$ The radius travels at the rate of 25 cm every second. What is the radius after 4 seconds? What is $A'(t)$ after 4 seconds?


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