Chapter 4.4, Problem 30E

To determine

To find: Dimensions of length and girth by given sum.

Answer to Problem 30E

The dimensions of lengths are $18\text{ in, 18 in}$ and the girth is $36\text{ in}$ .

Explanation of Solution

Given:

Sum of the length and the grith should not exceed $108\text{ in}$ .

Calculation:

Let the length of the side of square base is $x$ and $y$ is the girth.

Then, it requires,

  $4x+y\le 108$

Assuming,

  $\begin{array}{l}4x+y=108\\ y=108-4x\end{array}$

The volume is

  $\begin{array}{l}V=(x^2)(108-4x)\\ V=108x^2-4x^3\end{array}$

For any function, $f(x)$ the condition of minimum and maximum is obtained by finding

  $f\text{'}(x)=0$ , critical points and end points.

  $\begin{array}{c}V=108x^2-4x^3,0<x<27\\ \frac{dV}{dx}=216x-12x^2\end{array}$

Critical points occur at

  $\begin{array}{c}\frac{dV}{dx}=0\\ 216x-12x^2=0\\ x(216-12x)=0\\ x=18\end{array}$

Critical point occurs at $x=18$ .

  $\frac{dV}{dx}>0$

For $0<x<18$ and $\frac{dV}{dx}>0$ for $18<r<27$ this corresponds to maximum volume.

For $x=18$

  $\begin{array}{l}h=108-4x\\ h=108-72\\ h=36\end{array}$

Thus,for maximum volume the dimensions of length are $18\text{ in, 18 in}$ and the girth is $36\text{ in}$ .