Answered

Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Connect with a community of experts ready to help you find solutions to your questions quickly and accurately. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.

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]

Sagot :

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]
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.