Answer
(a) $\frac{dA}{dr} = 8~\pi$ ft^2/ft
(b) $\frac{dA}{dr} = 16~\pi$ ft^2/ft
(c) $\frac{dA}{dr} = 24~\pi$ ft^2/ft
The rate of change of the surface area increases linearly as the radius increases.
Work Step by Step
$S = 4\pi~r^2$
We can find the rate of change of the area:
$\frac{dA}{dr} = (8~\pi)~r$
(a) We can find the rate at which the area is increasing when $r = 1~ft$:
$\frac{dA}{dr} = (8~\pi)~r$
$\frac{dA}{dr} = (8~\pi)~(1)$
$\frac{dA}{dr} = 8~\pi$
(b) We can find the rate at which the area is increasing when $r = 2~ft$:
$\frac{dA}{dr} = (8~\pi)~r$
$\frac{dA}{dr} = (8~\pi)~(2)$
$\frac{dA}{dr} = 16~\pi$
(c) We can find the rate at which the area is increasing when $r = 3~ft$:
$\frac{dA}{dr} = (8~\pi)~r$
$\frac{dA}{dr} = (8~\pi)~(3)$
$\frac{dA}{dr} = 24~\pi$
The rate of change of the surface area increases linearly as the radius increases.
