Answer
(a) $\frac{dA}{dt} = (7200~\pi)~cm^2/s$
(b) $\frac{dA}{dt} = (21,600~\pi)~cm^2/s$
(c) $\frac{dA}{dt} = (36,000~\pi)~cm^2/s$
The rate of change of the area increases linearly as time increases.
Work Step by Step
Note that $r(t) = 60t$
We can write a function for the area:
$A = \pi~r^2$
$A = \pi~(60t)^2$
$A = (3600~\pi)~t^2$
We can find the rate of change of the area:
$\frac{dA}{dt} = (7200~\pi)~t$
(a) We can find the rate at which the area is increasing after $1~s$:
$\frac{dA}{dt} = (7200~\pi)~t$
$\frac{dA}{dt} = (7200~\pi)~(1)$
$\frac{dA}{dt} = (7200~\pi)~cm^2/s$
(b) We can find the rate at which the area is increasing after $3~s$:
$\frac{dA}{dt} = (7200~\pi)~t$
$\frac{dA}{dt} = (7200~\pi)~(3)$
$\frac{dA}{dt} = (21,600~\pi)~cm^2/s$
(c) We can find the rate at which the area is increasing after $5~s$:
$\frac{dA}{dt} = (7200~\pi)~t$
$\frac{dA}{dt} = (7200~\pi)~(5)$
$\frac{dA}{dt} = (36,000~\pi)~cm^2/s$
The rate of change of the area increases linearly as time increases.
