Question

A stone is thrown into a pond. A circular ripple is spreading over the pond in such a way that the radius is increasing at the rate of 3.3 feet per second. Find a function, r(t), for the radius in terms of t. Find a function, A(r), for the area of the ripple in terms of r. Find (A o r)(t).

Answer 1

To find the function for the radius, we know that the rate of change of the radius is given as 3.3 feet per second. We can integrate this to get the function for the radius:

∫ dr/dt dt = ∫ 3.3 dt

r = 3.3t + C

We know that when t = 0, r = 0. So, we can solve for C as follows:

0 = 3.3(0) + C

C = 0

Therefore, the function for the radius is:

r(t) = 3.3t

To find the function for the area of the ripple in terms of r, we use the formula for the area of a circle:

A = πr^2

Substituting r = 3.3t, we get:

A(r) = π(3.3t)^2

A(r) = 34.56πt^2

Finally, to find (A o r)(t), we substitute r(t) into A(r) and get:

(A o r)(t) = A(r(t))

(A o r)(t) = A(3.3t)

(A o r)(t) = 34.56πt^2