Question

question: a pebble is dropped into a calm pond, causing ripples in the form of concentric circles. the radius of the outer ripple is increasing at a constant rate of 6 inches per second. when the radius is 4 feet, at what rate (in ft²/sec) is the total area a of the disturbed water changing? round your answer to two decimal places. provide your answer below.

Explanation:

Step1: Recall the area formula for a circle

The area formula of a circle is $A = \pi r^{2}$, where $A$ is the area and $r$ is the radius.

Step2: Differentiate the area formula with respect to time $t$

Using the chain - rule, $\frac{dA}{dt}=2\pi r\frac{dr}{dt}$.

Step3: Convert the rate of change of radius to feet per second

Given $\frac{dr}{dt}=6$ inches per second. Since 1 foot = 12 inches, $\frac{dr}{dt}=\frac{6}{12}=0.5$ feet per second.

Step4: Substitute the values of $r$ and $\frac{dr}{dt}$ into the derivative formula

We know $r = 4$ feet and $\frac{dr}{dt}=0.5$ feet per second. Substituting into $\frac{dA}{dt}=2\pi r\frac{dr}{dt}$, we get $\frac{dA}{dt}=2\pi\times4\times0.5$.

Step5: Calculate the value of $\frac{dA}{dt}$

$\frac{dA}{dt}=4\pi\approx 12.57$ square feet per second.

Answer:

$12.57$