A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 25 cm/s.
If A is the area of this circle as a function of the radius, find A composed r.

To solve this problem, we need to find the area \( A \) of the circular ripple as a function of time, where the radius of the ripple changes with time.

### Step 1: Find \( A \) as a function of the radius \( r \)
The area \( A \) of a circle is given by the formula:
\[
A = \pi r^2
\]
This is the function \( A(r) \), where \( r \) is the radius of the ripple at any given moment.

### Step 2: Express the radius \( r \) as a function of time \( t \)
We know that the ripple travels outward at a speed of 25 cm/s. The radius \( r \) of the ripple increases linearly with time. Therefore, we can express \( r \) as:
\[
r(t) = 25t
\]
where \( t \) is the time in seconds, and \( r(t) \) is the radius after \( t \) seconds.

### Step 3: Compose \( A \) with \( r(t) \)
Now we want to find the area \( A \) as a function of time, so we substitute \( r(t) = 25t \) into the equation for \( A(r) \):
\[
A(r(t)) = \pi (r(t))^2 = \pi (25t)^2 = \pi \cdot 625t^2
\]
Thus, the area of the circular ripple as a function of time is:
\[
A(t) = 625\pi t^2
\]

### Final Answer:
The area of the circular ripple as a function of time is:
\[
A(t) = 625\pi t^2
\]

Would you like more details on any part of this solution?

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Here are 5 related questions:
1. How would the result change if the speed of the ripple was different?
2. What is the rate of change of the area with respect to time?
3. How would you compute the circumference of the ripple as a function of time?
4. What would the equation be if the ripple slowed down over time?
5. How do you find the derivative of the area function \( A(t) \)?

**Tip:** To check if a composed function is correct, verify both individual functions first before substituting!

A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 25 cm/s. If A is the area of this circle as a function of the radius, find A composed r.

Explore how to find the area of a circular ripple in a lake as a function of time. This detailed solution covers key geometric concepts, function composition, and rate of change calculations, suitable for high school students in Grades 9-12.

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