Answered

The radius of a circular pond is increasing at a constant rate, which can be modeled by the function [tex]r(t) = 5t[/tex], where [tex]t[/tex] is time in months. The area of the pond is modeled by the function [tex]A(r) = \pi r^2[/tex]. The area of the pond with respect to time can be modeled by the composition [tex]A(r(t))[/tex].

Which function represents the area with respect to time?

A. [tex]A(r(t)) = 5 \pi r^2[/tex]
B. [tex]A(\tau(t)) = 10 \approx \tau^2[/tex]
C. [tex]A(r(t)) = 25 \pi t^2[/tex]
D. [tex]A(r(t)) = 5 \pi t^2[/tex]



Answer :

GinnyAnswer
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]

Other Questions