Answer
(a) $S'(1)$ = $8{\pi}$ $ft^{2}/ft$
(b) $S'(2)$ = $16{\pi}$ $ft^{2}/ft$
(c) $S'(3)$ = $24{\pi}$ $ft^{2}/ft$
As the radius increases, the surface area grows at an increasing rate. In fact, the rate of change is linear with respect to the radius.
Work Step by Step
$S(r)$ = $4{\pi}r^{2}$
$S'(r)$ = $8{\pi}r$
(a)
$S'(1)$ = $8{\pi}(1)$ = $8{\pi}$ $ft^{2}/ft$
(b)
$S'(2)$ = $8{\pi}(2)$ = $16{\pi}$ $ft^{2}/ft$
(c)
$S'(3)$ = $8{\pi}(3)$ = $24{\pi}$ $ft^{2}/ft$
As the radius increases, the surface area grows at an increasing rate. In fact, the rate of change is linear with respect to the radius.