Question

A large plywood box has a volume of. Its length is 9 ft greater than its height, and its width is 4 ft less than its height. What are the dimensions of the box.

Short answer

The dimension of the box is $2\text{ft}\times \text{6ft}\times \text{15ft}$.

Step-by-step solution

Step-1. Volume of box.

The volume of the box is$V=lwh$ where, l, w, h are the length, width, and height respectively.

Step-2. Substitute the respective values.

It is given that the volume is $180{\text{ft}}^3$.

Let the height of the box is x ft.

Then the length of the box is$x+9$ ft and its width is$x-4$ ft.

Substitute the respective values into the volume of the box formula.

$180=(x-4)\left(x\right)(x+9)$

Step-3. Simplify the equation.

$\begin{array}{l}180=(x-4)\left(x\right)(x+9)\\ x^3-4x^2+9x^2-36x=180\\ x^3+5x^2-36x-180=0\\ x^2(x+5)-36(x+5)=0\\ (x^2-36)(x+5)=0\\ (x+6)(x-6)(x+5)=0\end{array}$

Therefore,

$x=-6,x=6,\text{or}x=-5$

Side should a positive number so ignore the negative values. So, the height of the box is 6 ft.

Then the length of the box is$6+9=15$ ft and its width is$6-4=2$ ft.

Hence, the dimension of the box is $2\text{ft}\times \text{6ft}\times \text{15ft}$.