Chapter 8.3, Problem 59SR

To determine

To find the dimension of the rectangular prism.

Answer to Problem 59SR

The dimensions of the rectangular prism is $\text{4cm, 8cm and 2cm}$ .

Explanation of Solution

Given information:

The given statement is:

“The width of a rectangular prism is $w$ centimeters. The height is $2$ centimeters less than the width. The length is $4$ centimeters more than the width. If the volume of the prism is $8$ times the measure of the length, find the dimensions of the prism.”

Formula used:

The following formula is used:

  $\text{Volume of a rectangle}=l\times b\times h$

Calculation:

Take the width of the rectangle be $w$ centimeters.

The height of the rectangle is $w-2$ centimeters and its length is $w+4$ centimeters.

The volume of the rectangle is $8\left(w+4\right)$ centimeters.

The dimensions can be calculated by:

  $\begin{array}{l}\left(w\right)\left(w+4\right)\left(w-2\right)=8\left(w+4\right)\\ \left(w^2+4w\right)\left(w-2\right)=8w+32\\ w^3-2w^2+4w^2-8w=8w+32\\ w^3+2w^2-8w-8w-32=0\\ w^3+2w^2-16w-32=0\\ \left(w^3+2w^2\right)+\left(-16w-32\right)=0\\ w^2\left(w+2\right)-16\left(w+2\right)=0\\ \left(w^2-16\right)\left(w+2\right)=0\\ \left(w-4\right)\left(w+4\right)\left(w+2\right)=0\\ w=4,-4\text{ or }-2\end{array}$

Since, the dimension of a rectangle cannot be negative.

Hence, the dimensions of the rectangular prism is $\text{4cm, 8cm and 2cm}$ .