Q 40. - Calculus

Question

suppose the sides of a cube are expanding at a rate of $2$ inches per minute. 

How fast is the volume of the cube changing at the moment that the cube has a side length of $20$ inches? 

Step-by-Step Solution

Answer

The rate of change in volume is $2400 i{n}^3/min$.

Step 1. Given Information

It is given that

$\frac{ds}{dt}=2 in/min, s=20 in$

Step 2. Setting up the rate of change of the volume equation

The formula for the volume of a cube is

$V=s^3$

where 

V is the volume of a cube 

s is the side length

Find derivative with respect to t

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$

Step 3. Finding the rate of change in Volume

Plug values $\frac{ds}{dt}=2 in/min , s=20 in$

$$\begin{gathered} \frac{dV}{dt}=3{\left(20\right)}^2\times 2 \\ \frac{dV}{dt}=2400 i{n}^3/min \end{gathered}$$