(a) The radius of the circle is the distance the wave travels since it first formed, so if g(t) is the radius of the circle at time t, then it changes at a rate according to
dg/dt = 60 cm/s
Integrate both sides with respect to t to solve for g :
∫ dg/dt dt = ∫ (60 cm/s) dt
g(t) = (60 cm/s) t + C
but C = 0 since the radius at t = 0 must be 0.
g(t) = (60 cm/s) t
(b) The area of any circle with radius r is πr ². So
f(r) = πr ²
(c) The composition of f with g represents the area of water encircled by the wave at time t :
(f o g)(t) = f(g(t)) = π g(t) ²
