College Algebra, Chapter 1, 1.5, Section 1.5, Problem 82

Suppose that a large plywood box has a volume of $180 ft^3$. Its length is $9 ft$ greater than its height, and its width is $4ft$ less than its height. Find the dimensions of the box.

Recall that the volume of the box is


$
\begin{equation}
\begin{aligned}

V =& L x W x H,
&&
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\text{ where } L =& 9 + W
&&
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W =& H - 4
&&
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V =& (9 + W)(H - 4)(H)
&& \text{Substitute } W = H - 4
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V =& (9 + (H - 4))(H - 4)(H)
&&
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V =& (H + 5)(H - 4)(H)
&& \text{Expand}
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V =& (H^2 + 5H - 20)H
&& \text{Substitute the given}
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180 =& H^3 + 5H^2 - 20H
&& \text{Subtract } 180
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0 =& H^3 + 5H^2 - 20H - 180
&& \text{Factor}
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0 =& (H + 6)(H - 6)(H + 5)
&& \text{Zero Product Property}
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H + 6 =& 0, H - 6 = 0 \text{ and } H + 5 = 0
&& \text{Solve for } H
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H =& -6, H = 6 \text{ and } H = -5
&& \text{Choose } H > 0
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\text{So if } H =& 6 ft, \text{ then}
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W =& H - 4 = 6- 4 = 2 ft
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L =& 9 + W = 9 + 2 = 11 ft
\end{aligned}
\end{equation}
$