Q 37. - Calculus

Question

A rock dropped into a pond causes a circular wave of ripples whose radius increases at $4$ inches per second. How fast is the area of the circle of ripples expanding at the instant that the radius of the circle is $12$ inches? $24$ inches? $100$ inches? Explain why it makes sense that the rate of change of the area increases as the radius increases. 

Step-by-Step Solution

Answer

For $r=12$inches: The rate of the area of the circle of ripples expanding at the instant   is $96\pi i{n}^2/sec$

For $r=24$inches: The rate of the area of the circle of ripples expanding at the instant   is $192\pi i{n}^2/sec$

For $r=100$ inches: The rate of the area of the circle of ripples expanding at the instant   is  $800\pi i{n}^2/sec$.

The rate of change of the area increases as the value of radius increases because circle is expanding.

Step 1. Given Information

It is given that 

$$\begin{gathered} \frac{dr}{dt}=4 in/sec \\ r=12 in, r=24 in, r=100 in \end{gathered}$$

Step 2. Setting up the rate of change of the area equation

The formula for the area of a circle is 

$A={\mathrm{\pi r}}^2$

Find derivative with respect to $t$ using chain rule

$$\begin{gathered} \frac{dA}{dt}=\pi \left(2r\right)\frac{dr}{dt} \\ \frac{dA}{dt}=2\pi r\frac{dr}{dt} \end{gathered}$$

Step 3. Finding the rate of change of area value

(I)

It is given that

$$\begin{gathered} r=12 in \\ \frac{dr}{dt}=4 in/sec \end{gathered}$$

Plug these values into rate of change of area formula

$$\begin{gathered} \frac{dA}{dt}=2\pi \left(12\right)\times 4 \\ \frac{dA}{dt}=96\pi i{n}^2/sec \end{gathered}$$

(II)

It is given that

$r=24 in, \frac{dr}{dt}=4 in/sec$

Plug these values into rate of change of area formula

(III)

It is given that

$r=100 in, \frac{dr}{dt}=6 in/sec$

Plug these values into rate of change of area formula

$$\begin{gathered} \frac{dA}{dt}=2\pi \left(100\right)\times 4 \\ \frac{dA}{dt}=800\pi i{n}^2/sec \end{gathered}$$

The rate of change of area is increasing because as radius increases ripple area will increase.