To determine how fast the area of the circular ripple is growing when the radius is 15 cm, let's follow these steps:
1. Understand the given values:
- The radius of the ripple is increasing at a rate of 3.5 cm/sec.
- The radius of the ripple at a particular moment is 15 cm.
2. Formula for the area of a circle:
- The area, [tex]\( A \)[/tex], of a circle is given by the formula:
[tex]\[
A = \pi r^2
\][/tex]
where [tex]\( r \)[/tex] is the radius.
3. Rate of change of the area with respect to time:
- To find how fast the area is changing with respect to time, we need the derivative of the area with respect to time, [tex]\( \frac{dA}{dt} \)[/tex].
4. Using the chain rule:
- The chain rule allows us to express [tex]\( \frac{dA}{dt} \)[/tex] in terms of [tex]\( \frac{dA}{dr} \)[/tex] and [tex]\( \frac{dr}{dt} \)[/tex].
[tex]\[
\frac{dA}{dt} = \frac{dA}{dr} \cdot \frac{dr}{dt}
\][/tex]
5. Find [tex]\( \frac{dA}{dr} \)[/tex]:
- First, find the derivative of the area with respect to the radius:
[tex]\[
\frac{dA}{dr} = \frac{d}{dr} (\pi r^2) = 2 \pi r
\][/tex]
6. Calculate [tex]\( \frac{dA}{dr} \)[/tex] at [tex]\( r = 15 \)[/tex] cm:
- Plugging in [tex]\( r = 15 \)[/tex] cm:
[tex]\[
\frac{dA}{dr} = 2 \pi \cdot 15 = 30 \pi
\][/tex]
7. Given [tex]\( \frac{dr}{dt} = 3.5 \)[/tex] cm/sec:
- Now, we multiply [tex]\( \frac{dA}{dr} \)[/tex] by [tex]\( \frac{dr}{dt} \)[/tex] to find [tex]\( \frac{dA}{dt} \)[/tex].
[tex]\[
\frac{dA}{dt} = 30 \pi \cdot 3.5
\][/tex]
8. Calculate [tex]\( \frac{dA}{dt} \)[/tex]:
- Compute the numerical value:
[tex]\[
\frac{dA}{dt} = 30 \pi \cdot 3.5 \approx 94.247 \cdot 3.5 \approx 329.867
\][/tex]
Therefore, the area of the ripple is growing at approximately 329.87 square centimeters per second when the radius is 15 cm.
