Throwing a Pebble

Calculus Level 2

A pebble is thrown into a still pound. The ripples caused by the disturbance form concentric circles that expand from the point where the pebble hits the water. What is the instantaneous rate that the area of one such circle is expanding when its radius is increasing at a rate of 0.5 m/s and it is currently having a radius of 4 m?

Problem from the app "Calculus Math App".

8 π m 2 / s 8\pi\ m^2/s 2 π m 2 / s 2\pi\ m^2/s 1 π m 2 / s 1\pi\ m^2/s 4 π m 2 / s 4\pi\ m^2/s

Victor Paes Plinio by Victor Paes Plinio , 21, Brazil

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1 solution

Tom Engelsman
Aug 1, 2017

Let A ( r ) = π r 2 A(r) = \pi r^2 be the area of the circular ripple. We will require the following related rates equation:

d A d t = d A d r ⋅ d r d t = ( 2 π r ) ( 0.5 m / s ) \frac{dA}{dt} = \frac{dA}{dr} \cdot \frac{dr}{dt} = (2\pi r)(0.5 m/s)

At r = 4 m r = 4m , we have d A d t = 2 π ( 4 ) ( 0.5 ) = 4 π m 2 / s \frac{dA}{dt} = 2\pi (4)(0.5) = \boxed{4\pi m^{2}/s}

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