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?

$15 cm^3/s$

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 = $a^3$

$⇒\frac{dV}{dt}=3a^2\frac{da}{dt}=3a^2×\frac{1}{2a}$

$=\frac{3}{2}a$

$∴\left.\frac{dV}{dt}\right|_{a=10}=\frac{3}{2}×10=15cm^3/s$