A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius (in meters) of the outer ripple is given by r(t) =0.2t, where t is the time in seconds after the pebble strikes the water. The area of the circle is given by the function A(r) = π . Find and interpret (Aₒr)(t).

Question

A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius (in meters) of the outer ripple is given by r(t) =0.2t, where t is the time in seconds after the pebble strikes the water. The area of the circle is given by the function A(r) = π . Find and interpret (Aₒr)(t).

Answer 1

Since A(r)=π is constant, A(r) is also constant.

Now, if you meant to say that A(r) = πr^2 then we have
(A◦r)(t) = A(r(t)) = π(0.2t)^2 = 0.04πt^2

Answer 2

To find the function (Aₒr)(t), we need to find the composition of the functions A(r) and r(t).

The function A(r) gives the area of a circle with radius r. So, substituting r(t) into A(r), we get:

A(r(t)) = π * (0.2t)^2

Simplifying this, we have:

A(r(t)) = π * 0.04t^2

Now, we need to find (Aₒr)(t), which is the composition of A(r) and r(t).

(Aₒr)(t) = A(r(t))

Substituting in the expression for A(r(t)) we found earlier:

(Aₒr)(t) = π * 0.04t^2

This is the final expression for (Aₒr)(t).

Interpretation:
The function (Aₒr)(t) represents the area of the circle formed by the outermost ripple in the pond at a given time t after the pebble is dropped. The area is given in terms of π and the square of the time t. As time increases, the ripple spreads out, causing the area of the circle to increase.