Chapter 2, Problem 74RE

(a)

To determine

To express: the amount $A$ of material used to make such a box as a function of the length $x$ of a side of the square base.

Answer to Problem 74RE

The function is $A\left(x\right)=2x^2+\frac{40}{x}$ .

Explanation of Solution

Given information:

A closed box with a square base is required to have a volume of 10 cubic feet.

Calculation:

Let $x$ be the length of side of the square base of the box, Let $h$ be the height of the box.

Volume of the box,

  $\begin{array}{c}V=\text{base}\text{area}\times \text{height}\\ =x^{2}h\\ h=\frac{V}{x^2}\end{array}$

Since $V=10$ ,

  $h=\frac{10}{x^2}$

Now, the amount $A$ of material used to make such a box is

  $\begin{array}{c}A=2\times \text{area}\text{of}\text{the}\text{base}+4\times \text{area}\text{of}\text{the}\text{side}\\ =2x^2+4xh\end{array}$

Since, $h=\frac{10}{x^2}$ ,

  $\begin{array}{c}A\left(x\right)=2x^2+4\left(\frac{10}{x^2}\right)x\\ =2x^2+\frac{40}{x}\end{array}$

So, the function is $A\left(x\right)=2x^2+\frac{40}{x}$ .

(b)

To determine

To find: the quantity of material required for a base 1 foot by 1 foot.

Answer to Problem 74RE

Required material is $42{\text{foot}}^{\text{2}}$ .

Explanation of Solution

Given information:

A closed box with a square base is required to have a volume of 10 cubic feet.

Calculation:

When $x=1$ ,

  $\begin{array}{c}A\left(1\right)=2{\left(1\right)}^2+\frac{40}{1}\\ =2+40\\ =42{\text{foot}}^{\text{2}}\end{array}$

So, required material is $42{\text{foot}}^{\text{2}}$ .

(c)

To determine

To find: the quantity of material required for a base 2 foot by 2 foot.

Answer to Problem 74RE

Required material is ${\text{28feet}}^{\text{2}}$ .

Explanation of Solution

Given information:

A closed box with a square base is required to have a volume of 10 cubic feet.

Calculation:

When $x=2$ ,

  $\begin{array}{c}A\left(2\right)=2{\left(2\right)}^2+\frac{40}{2}\\ =8+20\\ =28{\text{feet}}^{\text{2}}\end{array}$

So, required material is ${\text{28feet}}^{\text{2}}$ .

(d)

To determine

To sketch: the graph of $A=A\left(x\right)$ , and to find the value of $x$ is $A$ smallest.

Answer to Problem 74RE

The minimum value of $A$ . The minimum is $27.85{\text{foot}}^{\text{2}}$ at $x=2.15\text{feet}$ .

Explanation of Solution

Given information:

A closed box with a square base is required to have a volume of 10 cubic feet.

Calculation:

The graph of the function $A\left(x\right)=2x^2+\frac{40}{x}$ is plotted below using Maple.

  

Using graphing software the minimum value of $A$ . The minimum is $27.85{\text{foot}}^{\text{2}}$ at $x=2.15\text{feet}$ .