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A rock is dropped Into a calm pond, causing ripples to expand in the form of circles. The radios of the putter ripple is increasing at a rate of 2 feet per second. When the radios is 6 feet what is the rate of change of the total area of the circle?

A rock is dropped Into a calm pond, causing ripples to expand in the form of circles-example-1
User Nurlan
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1 Answer

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First, let's define a function fo the radius in term of the time, in secons, since the rock hit the pond:


r(t)=2t

Now, we know that the area of the circle, in terms of the radius is:


A(r)=\pi r^2

But since we know that the radius is a function of time,


\begin{gathered} A(r(t))=\pi(2t)^2 \\ \\ \Rightarrow(A\circ r)(t)=4\pi t^2 \end{gathered}

This way, we'll have that:


(A\circ t)^(\prime)(t)=8\pi t

Now, we find the time at which the radius was 6 feet:


\begin{gathered} 6=2t\rightarrow(6)/(2)=t \\ \Rightarrow t=3 \end{gathered}

Now,


\begin{gathered} (A\circ t)^(\prime)(3)=8\pi(3) \\ \\ \rightarrow(A\circ t)^(\prime)(3)=75.40 \end{gathered}

Therefore, we can conclude that the rate of change of the total area of the circle when the radius is 6 feet is 75.40 square feet per second.

User Nandhini
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