a stone is thrown into a pond, creating a circular ripple that spreads over the pond in such a way that the…

a stone is thrown into a pond, creating a circular ripple that spreads over the pond in such a way that the radius is increasing at a rate of 3.4 ft/sec. the function for the radius in terms of t is r(t)=3.4t. the function a(r) for the area of the ripple in terms of the radius r is a(r)=πr². find and interpret a(r(t)). a(r(t))= (simplify your answer. type an exact answer in terms of π. use integers or decimals for any numbers in the expression )

a stone is thrown into a pond, creating a circular ripple that spreads over the pond in such a way that the radius is increasing at a rate of 3.4 ft/sec. the function for the radius in terms of t is r(t)=3.4t. the function a(r) for the area of the ripple in terms of the radius r is a(r)=πr². find and interpret a(r(t)). a(r(t))= (simplify your answer. type an exact answer in terms of π. use integers or decimals for any numbers in the expression )

Answer

Explanation:

Step1: Recall function - composition concept

We know that (A(r)=\pi r^{2}) and (r(t) = 3.4t). To find (A(r(t))), we substitute (r(t)) into (A(r)).

Step2: Substitute (r = r(t)) into (A(r))

Substitute (r=3.4t) into (A(r)=\pi r^{2}), we get (A(r(t))=\pi(3.4t)^{2}).

Step3: Simplify the expression

((3.4t)^{2}=3.4^{2}t^{2}=11.56t^{2}), so (A(r(t)) = 11.56\pi t^{2}).

Answer:

(11.56\pi t^{2})