Question

# A packing company is doing an inventory of boxes. Their most popular box is display below. You can use the formula V = lwh to find the volume of the b

Solid Geometry

A packing company is doing an inventory of boxes. Their most popular box is display below.
You can use the formula $$V = lwh$$ to find the volume of the box.
The volume of the box is $$40 ft^{3}$$. What is the value of x? Find the length and the width of the box. Describe any extraneous solutions.

2021-05-08

From the figure, $$l=3x−5 ft, w=2x−1 ft,$$ and $$h=2 ft$$. The volume is $$40 ft^3$$ so we can write:
$$V=lwh$$
$$40=(3x−5)(2x−1)(2)$$
Divide both sides by 2:
$$20=(3x−5)(2x−1)$$
Expand the right side:
$$\displaystyle{20}={6}{x}^{{2}}−{13}{x}+{5}$$
Write in standard form:
$$\displaystyle{0}={6}{x}^{{2}}−{13}{x}−{15}$$
Factor the right side:
$$0=(6x+5)(x−3)$$
By zero-product property,
$$x=−56,3$$
Since $$x=−56$$ gives negative length and width, then it is extraneous. Hence,
$$x=3$$
Using the value of $$xx$$, the length is $$3(3)−5=4$$ ft ft and the width is $$2(3)−1=5 ft.$$