The surface area of a cube increases at a rate of 6 square centimeters per second. How fast is the volume increasing when the length of an edge is 10 centimeters?
Answer & explanation
Correct answer: option 1
A, surface area of cube = $6a^2$
$⇒\frac{dA}{dt}=12a\frac{da}{dt}$
$⇒6=12a\frac{da}{dt}$
$⇒\frac{da}{dt}=\frac{1}{2a}$
V, Volume = $a^3$
$⇒\frac{dV}{dt}=3a^2\frac{da}{dt}=3a^2×\frac{1}{2a}$
$=\frac{3}{2}a$
$∴\left.\frac{dV}{dt}\right|_{a=10}=\frac{3}{2}×10=15cm^3/s$