To solve this, we need to find the area of the pond with respect to time, given the models for the radius and the area.
1. Understand the given functions:
- The radius of the pond as a function of time is [tex]\( r(t) = 5t \)[/tex].
- The area of the pond as a function of its radius is [tex]\( A(r) = \pi r^2 \)[/tex].
2. Find the composition of these functions to get the area in terms of time:
- We need to substitute the radius function [tex]\( r(t) = 5t \)[/tex] into the area function [tex]\( A(r) = \pi r^2 \)[/tex].
3. Substitute [tex]\( r(t) \)[/tex] into [tex]\( A(r) \)[/tex]:
- First, replace [tex]\( r \)[/tex] with [tex]\( 5t \)[/tex] in the area function.
- [tex]\( A(r(t)) = \pi (r(t))^2 \)[/tex].
4. Calculate [tex]\( A(r(t)) \)[/tex]:
- Substitute [tex]\( r(t) = 5t \)[/tex]:
[tex]\[
A(r(t)) = \pi (5t)^2
\][/tex]
- Simplify the expression:
[tex]\[
A(r(t)) = \pi \cdot 25t^2
\][/tex]
- Thus,
[tex]\[
A(r(t)) = 25\pi t^2
\][/tex]
5. Identify the correct option:
- The function that correctly represents the area of the pond with respect to time is [tex]\( A(r(t)) = 25\pi t^2 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{D. \ A(r(t)) = 25 \pi t^2} \][/tex]
