Question

All the edges of a cube are expanding at a rate of 4 in. per second. How fast is the volume changing when each edge is 10 in. long

Answer 1

Given:

All the edges of a cube are expanding at a rate of 4 in. per second.

To find:

The rate of change in volume when each edge is 10 in. long.

Solution:

Let a be the edge of the cube.

According to the question, we get

$\dfrac{da}{dt}=4\text{ in./sec}$

$a=10\text{ in.}$

We know that, volume of a cube is

$V=a^3$

Differentiate with respect to t.

$\dfrac{dV}{dt}=\dfrac{d}{dt}a^3$

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

Putting the given values, we get

$\dfrac{dV}{dt}=(3(10)^2)\times 4$

$\dfrac{dV}{dt}=3(100)\times 4$

$\dfrac{dV}{dt}=300\times 4$

$\dfrac{dV}{dt}=1200\text{ in}^3\text{/sec}$

Therefore, the rate of change in volume 1200 cubic inches per second.