Question

example 4
pars wants to make a box by cutting out corners of a piece of paper and folding up the sides. the cardboard is 9 inches by 13 inches. the volume v(x) in cubic inches of the box is a function of the side lengths x in inches of the square cutouts.

1. write the expressions for v(x).
2. find v(4) and explain what it means in context.

Explanation:

Step1: Determine box dimensions

When we cut out squares of side - length $x$ from each corner of a cardboard that is 9 inches by 13 inches and fold up the sides, the length of the box $L=13 - 2x$, the width of the box $W = 9-2x$, and the height of the box $H=x$. The volume formula for a rectangular - box is $V = L\times W\times H$.

Step2: Write the volume function

Substitute the values of $L$, $W$, and $H$ into the volume formula. So, $V(x)=(13 - 2x)(9 - 2x)x$. Expand the first two factors: $(13 - 2x)(9 - 2x)=13\times9-13\times2x-2x\times9 + 4x^{2}=117-26x - 18x+4x^{2}=117 - 44x+4x^{2}$. Then $V(x)=(117 - 44x + 4x^{2})x=4x^{3}-44x^{2}+117x$.

Step3: Calculate $V(4)$

Substitute $x = 4$ into $V(x)$:
\[

$$\begin{align*} V(4)&=4\times4^{3}-44\times4^{2}+117\times4\\ &=4\times64-44\times16 + 468\\ &=256-704+468\\ &=20 \end{align*}$$

\]
In the context of the problem, $V(4)$ represents the volume of the box (in cubic inches) when the side - length of the square cutouts is 4 inches.

Answer:

1. $V(x)=4x^{3}-44x^{2}+117x$
2. $V(4) = 20$. It means the volume of the box is 20 cubic inches when the side - length of the square cutouts is 4 inches.