Chapter 6.8, Problem 24PT

To determine

To find: the dimensions of the prism using volume of the prism.

Answer to Problem 24PT

The dimension of the prism is $9\text{ cm}$ .

Explanation of Solution

Given:

  $\begin{array}{l}w,w-5,w+8.\\ \text{volume}=612.\end{array}$

Concept used:

Volume of rectangular prism:

  $V=l\times w\times h$ .

Distance cannot be negative.

Calculation:

The side or the dimension of the rectangular prism are:

  $\begin{array}{l}w,w-5,w+8.\\ \text{volume}=612.\end{array}$

Mensuration formula for volume of rectangular prism is:

  $\begin{array}{l}V=l\times w\times h\\ V=w\left(w-5\right)\left(w+8\right).\\ V=w^3+3w^2-40w-612=0.\end{array}$

The equation of the volume can be further solved as:

  $\begin{array}{l}V=w^3+3w^2-40w-612=0.\\ V=\left(w-9\right)\left(w^2+12w+68\right)=0.\\ w=9,w=-6-4\sqrt{2}i\text{ and }w=-6+4\sqrt{2}i.\end{array}$

Out of three values $w$ should take positive $w$ cannot be negative since, distance cannot be taken negative.

So, the real solution is $w=9$ .

Hence, the dimension of the prism is $9\text{ cm}$ .