The length of a side of a cube is increasing at a rate of 4 inches per second. At what rate is the surface area increasing when the volume of the cube is 216 square inches ?

Apr 12, 2018

288 inches per second

Explanation:

$V = {s}^{3}$ (volume of cube formula)

when $v = 216$, $s = {216}^{\frac{1}{3}} = 6$

$A = 6 {s}^{2}$ (surface area)

differentiate $A = 6 {s}^{2}$ with respect to $t$ (time):
$\frac{\mathrm{dA}}{\mathrm{dt}} = 12 s \frac{\mathrm{ds}}{\mathrm{dt}}$
$\frac{\mathrm{dA}}{\mathrm{dt}} = 12 \left(6\right) \left(4\right)$ (because side length increasing at 4 in/s)
$\frac{\mathrm{dA}}{\mathrm{dt}} = 288$ inches per second