 # A stone is dropped into a lake, creating a circular oliviayychengwh 2021-12-19 Answered

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 1 s.

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Step 1
The ripple travels outward at 60 cm/s, this is the rate of change of the radius with respect to time
$\frac{dr}{dt}=60c\frac{m}{s}$
From the problem description we can also get an equation that gives the radius at time t.
r=60t
Step 2
The area of a circle is $A=\pi {r}^{2}$ , and we need to find the rate that this is increasing, which is $\frac{dA}{dt}$. So differentiate the A equation with respect to time t.
$A=\pi {r}^{2}$
$\frac{dA}{dt}=\pi 2r\frac{dr}{dt}$
We know the value of $\frac{dr}{dt}$ so we can substitute that in. Furthermore we can substitute r=60 t so that we have an equation where all we need to do is plug in the time t to get the rate of change of the area.
$\frac{dA}{dt}=\pi 2\left(60t\right)\left(60\right)=7200\pi$
Step 3
So for t=1
$\frac{dA}{dt}=7200\pi \left(1\right)\approx 22619c\frac{{m}^{2}}{s}$

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