Q 42. - 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 area of the cube’s base is $10$ square inches? 

Step-by-Step Solution

Answer

The rate of change in volume is $60 i{n}^3/min$

Step 1. Given Information

It is given that

$$\begin{gathered} \frac{ds}{dt}=2 in/min \\ A=s^2=10 i{n}^2 \end{gathered}$$

Step 2. Finding the rate of change in Volume

The volume equation is 

$V=s^3$

Find derivative with respect to t 

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

Plug values $\frac{ds}{dt}=2 in/min, s^2=10$

$$\begin{gathered} \frac{dV}{dt}=3\left(10\right)\times 2 \\ \frac{dV}{dt}=60 i{n}^3/min \end{gathered}$$