Chapter 4, Problem 46RE

To determine

To find : The dimension of largest open top storage bin.

Answer to Problem 46RE

The answer:

Base is $6\text{ft by }6\text{f}$ and height is $3\text{ft}$ .

Explanation of Solution

Given information:

The vertical sides that can be made from $108{\text{ft}}^2$ of the sheet steel.

Calculation:

  $x=$ Length of a base side

  $h=$ height of bin

  $V=$ volume of bin

  $108=$ surface area of bin

  $\begin{array}{c}V=h{x}^2\\ 108=x^2+4hx\end{array}$

Obtain the volume equation with a single variable.

  $\begin{array}{c}4hx=108-x^2\\ \Rightarrow h=\frac{108-x^2}{4x}\\ =\frac{27}{x}-\frac{1}{4}x\\ V=\left(\frac{27}{x}-\frac{1}{4}x\right)x^2\\ =27x-\frac{1}{4}{x}^3\end{array}$

To find $V^{\prime }$ where it is $=0$

  $\begin{array}{c}{V}^{\prime }=27-\frac{3}{4}{x}^2\\ \Rightarrow 0=27-\frac{3}{4}{x}^2\\ \Rightarrow x^2=\frac{4}{3}\cdot 27\\ =36\\ \Rightarrow x=6\end{array}$

To find $h$ , re-enter it into the second equation above.

  $\begin{array}{c}h=\frac{108-x^2}{4x}\\ =\frac{108-{(6)}^2}{4(6)}\\ =3\end{array}$

Therefore, the required dimension of largest open top storage bin:

Base is $6\text{ft by }6\text{f}$ and height is $3\text{ft}$ .