Q. 25 - Precalculus Enhanced with Graphing Utilities

Question

An open box with a square base is to be made from a square piece of cardboard 24 inches on a side by cutting out a square from each corner and turning up the sides. See the figure.

(a) Express the volume V of the box as a function of the length x of the side of the square cut from each corner.

(b) What is the volume if a 3-inch square is cut out?

(c) What is the volume if a 10-inch square is cut out?

(d) Graph $V=V\left(x\right)$. For what value of x is V largest?

Step-by-Step Solution

Answer

Part (a) Volume of the box is $V\left(x\right)=x{(24-2x)}^2$.

Part (b) $V\left(3\right)=972{\mathrm{in}}^3$

Part (c) $V\left(10\right)=160{in}^3$

Part (d) V(x) is largest when x is $4$ inches and its graph is

Part (a). Step 1. Use the formula for the volume of a cuboid.

From the given figure,

Height of the cuboid $=x$

Breadth of the cuboid $=$width of the cuboid $=24-2x$

Volume of the box $$\begin{gathered} =(24-2x)(24-2x)x \\ =x{(24-2x)}^2 \end{gathered}$$

So, $V\left(x\right)=x{(24-2x)}^2$

Part (b) Step 1. Substitute x = 3 inches in V ( x ) = x ( 24 - 2 x ) 2

This gives

$$\begin{gathered} V\left(3\right)=3{(24-2\times 3)}^2 \\ V\left(3\right)=3{(24-6)}^2 \\ V\left(3\right)=3{\left(18\right)}^2 \\ V\left(3\right)=972{\mathrm{in}}^3 \end{gathered}$$

Part (c) Step 1. Substitute x = 10 inches in V ( x ) = x ( 24 - 2 x ) 2 .

This gives

$$\begin{gathered} V\left(10\right)=10{(24-2\times 10)}^2 \\ V\left(10\right)=10{\left(4\right)}^2 \\ V\left(10\right)=10\left(16\right) \\ V\left(10\right)=160{\mathrm{in}}^3 \end{gathered}$$

Part (d). Step 1. The graph of V ( x ) = x ( 24 - 2 x ) 2 is as follows.

From the graph, 

V(x) is largest when $x=4$ inches.