Question

The edge length of a cube is changing at a rate of 10 in/sec. At what rate is the cube's volume changing when the edge length is 3 inches?

Answer 1

Given:

The edge length of a cube is changing at a rate of 10 in/sec.

To find:

The rate by which cube's volume changing when the edge length is 3 inches.

Solution:

We have,

$\dfrac{da}{dt}=10\text{ in/sec}$

We know that, volume of cube is

$V=a^3$

Differentiate with respect to t.

$\dfrac{dV}{dt}=3a^2\dfrac{da}{dt}$

Substituting $\dfrac{da}{dt}=10$ and a=3, we get

$\dfrac{dV}{dt}_{a=3}=3(3)^2(10)$

$\dfrac{dV}{dt}_{a=3}=3(9)(10)$

$\dfrac{dV}{dt}_{a=3}=270$

Therefore, the volume increased by 270 cubic inches per sec.