the circumference of a circular ripple is growing at a rate of 10cm per second. Find the rate at which its area is increasing when its radius is 20cm

To solve this problem, we need to relate the rate of change of the circumference of the ripple to the rate of change of its area.

### Step 1: Relationship between Circumference and Radius
The formula for the circumference \( C \) of a circle is:
\[
C = 2\pi r
\]
where \( r \) is the radius of the circle.

Given that the circumference is growing at a rate of \( \frac{dC}{dt} = 10 \, \text{cm/s} \), we can differentiate \( C = 2\pi r \) with respect to time \( t \):
\[
\frac{dC}{dt} = 2\pi \frac{dr}{dt}
\]
Now, substitute the given value \( \frac{dC}{dt} = 10 \):
\[
10 = 2\pi \frac{dr}{dt}
\]
Solving for \( \frac{dr}{dt} \), we get:
\[
\frac{dr}{dt} = \frac{10}{2\pi} = \frac{5}{\pi} \, \text{cm/s}
\]

### Step 2: Relationship between Area and Radius
The formula for the area \( A \) of a circle is:
\[
A = \pi r^2
\]
Now, differentiate this with respect to time \( t \):
\[
\frac{dA}{dt} = 2\pi r \frac{dr}{dt}
\]

### Step 3: Substituting Known Values
We are asked to find the rate at which the area is increasing when the radius is \( r = 20 \, \text{cm} \). Substituting \( r = 20 \) and \( \frac{dr}{dt} = \frac{5}{\pi} \) into the equation for \( \frac{dA}{dt} \):
\[
\frac{dA}{dt} = 2\pi (20) \left(\frac{5}{\pi}\right)
\]
Simplifying:
\[
\frac{dA}{dt} = 2 \times 20 \times 5 = 200 \, \text{cm}^2/\text{s}
\]

### Final Answer:
The area of the circular ripple is increasing at a rate of \( 200 \, \text{cm}^2/\text{s} \) when its radius is 20 cm.

Would you like more details or have any questions?

Here are five related questions to expand on this:

1. How does the rate of area change vary as the radius increases?
2. What would be the rate of area increase if the radius was 30 cm?
3. How does the rate of change of the radius affect the rate of change of the area?
4. What if the circumference growth rate changed to 15 cm/s? How would that affect the area growth?
5. Can you derive the formula for the rate of change of the area directly from the rate of change of the circumference?

**Tip**: When dealing with related rates, always check how variables are interrelated through equations and differentiate accordingly!

The circumference of a circular ripple is growing at a rate of 10cm per second. Find the rate at which its area is increasing when its radius is 20cm.

Discover how to calculate the rate of area increase for a circular ripple growing at 10 cm/s when the radius is 20 cm. This problem covers key calculus concepts, including related rates and differentiation, suitable for high school students in Grades 10-12.

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