To determine the function that represents the area of the pond with respect to time, we need to compose the given functions [tex]\( r(t) \)[/tex] and [tex]\( A(r) \)[/tex].
1. We start with the function for the radius of the pond, which is given as [tex]\( r(t) = 5t \)[/tex], where [tex]\( t \)[/tex] is the time in months.
2. The area [tex]\( A(r) \)[/tex] of a circular pond is modeled by the function [tex]\( A(r) = \pi r^2 \)[/tex], where [tex]\( r \)[/tex] is the radius.
To find the area with respect to time, we need to substitute the radius function [tex]\( r(t) \)[/tex] into the area function [tex]\( A(r) \)[/tex]. This is the composition [tex]\( A(r(t)) \)[/tex].
3. Substitute [tex]\( r(t) = 5t \)[/tex] into [tex]\( A(r) \)[/tex]:
[tex]\[
A(r(t)) = A(5t) = \pi (5t)^2
\][/tex]
4. Simplify the expression:
[tex]\[
A(5t) = \pi (5t)^2 = \pi (25t^2) = 25\pi t^2
\][/tex]
Therefore, the function that represents the area with respect to time is:
[tex]\[
A(r(t)) = 25\pi t^2
\][/tex]
So, the correct choice is:
C. [tex]\( A(r(t)) = 25\pi t^2 \)[/tex]
