Chapter 2, Problem 11P

To determine

To find: The function that model surface area of rectangular box in terms of length $x$ of one side of its square base whose volume is $60{\text{ft}}^{\text{3}}$.

Answer to Problem 11P

The function that model surface area in terms of the length is $\frac{240}{x}+2x^2$.

Explanation of Solution

Surface area $\left(S\right)$ of rectangular box is,

$\text{S}=\text{Area}\text{of}\text{four}\text{sides}+\text{Area}\text{of}\text{top}\text{and}\text{bottom}$

Volume of a rectangular box is,

$\begin{array}{c}V=\left(\text{base}\text{area}\right)\cdot \left(\text{height}\right)\\ =A\cdot h\end{array}$

Length of the side of its base is $x$ and base is a square.

Area of square is,

$\begin{array}{c}A=\left(\text{Side}\right)\cdot \left(\text{Side}\right)\\ =x\cdot x\\ =x^2\end{array}$

Calculate $h$ in terms of $x$.

$V=A\cdot h$

Substitute 60 for $V$ in above equation and solve for $h$.

$\begin{array}{c}60=x^2\cdot h\\ \frac{60}{x^2}=h\end{array}$

Area of four sides is,

$\begin{array}{c}\text{Area}\text{of}\text{four}\text{sides}=4\cdot x\left(\frac{60}{x^2}\right)\\ =\frac{240}{x}\end{array}$

Area of top and bottom faces of rectangular box is,

$\text{Area}\text{of}\text{top}\text{and}\text{bottom}\text{faces}=2x^2$

Summarize all the information in a table as shown below.

In Words	In Algebra
Area of square base	$x^2$
Height	$\frac{60}{x^2}$
Area of four sides	$\frac{240}{x}$
Area of top and bottom faces	$2x^2$

Use the information in the table and model the function.

$\begin{array}{c}\text{S}=\text{Area}\text{of}\text{four}\text{sides}+\text{Area}\text{of}\text{top}\text{and}\text{bottom}\text{faces}\\ =\frac{240}{x}+2x^2\end{array}$

Thus, the function that model surface area in terms of the length is $\frac{240}{x}+2x^2$.