Chapter 5.6, Problem 39E

To determine

To find: the length of each edge of the cube.

Answer to Problem 39E

The length of each edge is E- $\text{8 inches}$

Explanation of Solution

Given information :

  $\frac{dv}{dt}=24\text{in}{\text{.}}^3\text{/min}\text{.}$

  $\frac{d\text{A}}{dt}={12}^2\text{in}\text{./min}\text{.}$

A is the surface area of the cube.

Assume all variables are differentiable function of t .

Calculation:

We have to find the length of each edge of the cube.

Therefore,

Surface area of a cube-

  $\text{A}=6l^2$

Differentiate with respect to t .

  $\begin{array}{l}\frac{d\text{A}}{dt}=6\cdot 2l\frac{dl}{dt}\\ \frac{d\text{A}}{dt}=12l\frac{dl}{dt}\end{array}$

Putting the value of $\frac{d\text{A}}{dt}=12$ in above equation,

  $\begin{array}{l}12=12l\frac{dl}{dt}\\ \frac{d\text{l}}{dt}=\frac{12}{12l}\\ \frac{d\text{l}}{dt}=\frac{1}{l}\end{array}$

Again,

Volume of a cube is

  $\text{V}=l^3$

Differentiate with respect to t .

  $\frac{dv}{dt}=3l^2\frac{dl}{dt}$

Putting the value of $\frac{dv}{dt}=24$ and $\frac{dl}{dt}=\frac{1}{l}$ in above equation,

  $\begin{array}{l}24=3l^2\left(\frac{1}{l}\right)\\ 24=3l\\ l=\frac{24}{3}\\ l=8\text{ inches}\end{array}$

Hence, the length of each edge is E- $\text{8 inches}$