Chapter 1.1, Problem 57E

To determine

To find: The function of the surface area of the open rectangular box with a square base in terms of the length of a side of the base and the domain of the function.

Answer to Problem 57E

The formula for the surface area of the open rectangular box as a function of the length of a side of the base is $S\left(a\right)=a^2+\frac{8}{a}$.

The domain of the surface area of the open rectangular box as a function of the length of a side of the base is $a>0$.

Explanation of Solution

Given:

The volume of the open rectangular box with a square base is 2 m³.

Formula used:

Volume of the rectangular box of length l, width b, height h is, $V=l\times b\times h$.

Surface area of the open rectangular box of length l, width b, height h is, $S=2\left(lh+wh\right)+lb$.

Calculation:

Let the length of the open rectangular box be a and the height of the open rectangular box is h.

Since the base of the open rectangular box is a square, the width of the box is same as the length. Therefore, the width of the open rectangular box is a.

Volume of the open rectangular box is, $V=a\times a\times h$.

That is, $V=a^{2}h$

Since the volume of the open rectangular box with a square base is 2 m³, the height of the box is computed as follows:

$\begin{array}{c}2=a^{2}h\\ h=\frac{2}{a^2}\end{array}$

Thus, the surface area of the open rectangular box is $S=2\left(ah+ah\right)+a^2$.

That is, $S=4ah+a^2$

Substitute the value of h in $S=4ah+a^2$ and simplify.

$\begin{array}{c}S=4a\left(\frac{2}{a^2}\right)+a^2\\ =\frac{8}{a}+a^2\end{array}$

Therefore, the formula for the surface area (S) of the open rectangular box as a function of the length of a side of the base (a) is $S\left(a\right)=a^2+\frac{8}{a}$.

The domain of the surface area of the open rectangular box as a function of the length of a side of the base is $a>0$.