Chapter 3.2, Problem 84E

Volume of a Box A cardboard box has a square base, with each edge of the base having length x inches, as shown in the figure. The total length of all 12 edges of the box is 144 in.

1. (a) Show that the volume of the box is given by the function $V(x)=\text{2}{x}^{\text{2}}(\text{18}-x)$.
2. (b) What is the domain of V? (Use the fact that length and volume must be positive.)
3. (c) Draw a graph of the function V and use it to estimate the maximum volume for such a box.

To determine

To show: The volume of the given box is $V(x)=2x^2\left(18-x\right)$.

Explanation of Solution

Proof:

The length of each edge of the base is x inches. The total length of all 12 edges of the box is 144 in.

The formula to calculate volume of the box is,

$\text{V}=Areaofthebase\times Heightofthebox$ (1)

Total length of all edges of the box is 144 in. Length of each edge of base is x and let height of edge is y. Then, the total length of all 12 edges is,

$\begin{array}{c}8\text{x}+4\text{y}=144\left(\begin{array}{l}Asthereare8edgesofbaseand\\ 4edgesofheight\end{array}\right)\\ 4\text{y}=144-8\text{x}\\ \text{y}=\frac{144-8\text{x}}{4}\\ \text{y}=36-2\text{x}\end{array}$

Area of the base is ${\text{x}}^2$ and the height of the box is y. Substitute ${\text{x}}^2$ for area of the base and y for the height of the box in equation (1) to calculate volume.

$\begin{array}{c}V=x^2\times y\\ =x^2\left(36-2x\right)\left[\because y=36-2x\right]\\ =36x^2-2x^3\\ =2x^2\left(18-x\right)\end{array}$

Hence, the volume of the box is given by the function $V\left(x\right)=2x^2\left(18-x\right)$.

To determine

To find: The domain of function V.

Explanation of Solution

The volume of the box exists only when the length of sides must be greater than the zero. So the domain of V is all real number for $x>0$.

Thus, the domain of function $V\left(x\right)=2x^2\left(18-x\right)$ is $D=\left\{x:x\in ℝ,x>0\right\}$.

To determine

To estimate: The maximum volume of the box and draw the graph.

Answer to Problem 84E

The maximum volume is $1728{\text{in}}^3$

Explanation of Solution

Maximum volume for the box means maximum value of the given function.

Zeros of the polynomial $2x^2\left(18-x\right)=0$ are $x=0$ and $x=18$. The y-intercept is $V\left(0\right)=0$. The table below gives the additional value of V(x).

x	V(x)
0	0
4	448
8	1280
12	1728
16	1024

Plot the points from above table and join them to make smooth curve.

Figure (1)

From the Figure (1), it is seen that the maximum value of V(x) at $x=12$ is 1728.

Thus, the maximum volume of the box is $1728{\text{in}}^3$.