Question

A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius (in feet) of the outermost ripple is given by r(t) = 0.6t, where t is time (in seconds) after the pebble strikes the water. The area of the circle is given by a(r) = πr2. Find (a ∘ r)(t).

Answer 1

Solution-

A pebble is dropped into a calm pond, causing ripples in the form of concentric circles.

The radius of the outer ripple is given by r(t)= 0.6t,

where t is the time in seconds after the pebble strikes the water.

The area of the circle is given by the function $a(r) = \pi r^2$

(a ◦ r)(t) = a(r(t))  ( ∵ Using function composition / plugging in the second function into the first function)

⇒$a(r(t))=a[0.6t]=(\pi)(0.6t)^2=0.36 \pi t^2$

∴ This is the expression for area of circle of outermost ripple, at time t after the pebble strikes the water.