Question

A jewelry box has a length that is 2 inches longer than the width and a height that is 1 inch smaller than the width. the volume of the box is 140 cubic inches. what is the width of the jewelry box?

Answer 1

Given that a jewelry box has a length that is 2 inches longer than the width and a height that is 1 inch smaller than the width.

Let the width of the box be w, then the length of the box is given by l = w + 2 and the height of the box is given by h = w - 1.

The volume of a box is given by

$V=l\times w\times h \\ \\ =w(w+2)(w-1) \\ \\ =w(w^2+w-2)=w^3+w^2-2w$

Given that the volume of the box is 140 cubic inches. Thus

$w^3+w^2-2w=140$

Thus, the width of the jewelry box is obtained as follows

$w^3+w^2-2w=140 \\ \\ \Rightarrow w^3+w^2-2w-140=0 \\ \\ \Rightarrow(w-5)(w^2+6w+28)=0 \\ \\ \Rightarrow w=5$

Therefore, the width of the jewelry box is 5 inches.

*Note that $w^2+6w+28=0$ gives complex roots, so we did not use it.