Chapter 5.8, Problem 2GP

To determine

To calculate: The dimensions of the prism,

The volume of a rectangular prism is $1056$ cubic centimeters. Length is $1$ cm more than the width, and height is $3$ cm less than the width.

Answer to Problem 2GP

The value of length, width and height of a rectangular prism is: $L=12cm,W=11cm,H=8cm$ .

Explanation of Solution

Given information:

The dimensions of the prism,

The volume of a rectangular prism is $1056$ cubic centimeters. Length is $1$ cm more than the width, and height is $3$ cm less than the width.

Formula used:

For volume of a rectangle is,

  $V=l\times w\times h$

Where:

  $\begin{array}{l}\text{length=l}\\ \text{width=w}\\ \text{height=h}\end{array}$ .

Calculation:

Consider the statement as:

The volume of a rectangular prism is $1056$ cubic centimeters. Length is $1$ cm more than the width, and height is $3$ cm less than the width.

The width of a prism is: $w\text{cm}$

Then the length is $1$ cm more than the width that means: $w+1$

The height is $3$ cm less than the width as: $w-3$

Recall that the area of the rectangle: $V=l\times w\times h$

Apply the values in the formulas:

  $\begin{array}{c}V=l\times w\times h\\ 1056=\left(w+1\right)\left(w\right)\left(w-3\right)\\ 1056=\left(w+1\right)\left(w^2-3w\right)\\ 1056=w^3-2w^2-3w\end{array}$

Simplified further:

  $\begin{array}{l}{w}^3-2w^2-3w=1056\\ w^3-2w^2-3w-1056=0\end{array}$

Now to find the list of positive rational solutions for the expression look at the factors of the constants.

That are:

  $\begin{array}{l}\pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 8,\pm 11,\pm 12,\pm 24,\pm 44,\\ \pm 88,\pm 96,\pm 132,\pm 176,\pm 352,\pm 528,\pm 1056\end{array}$

Here the last two terms subtracting values from the cube of the first and want to get zero,

So as have to take the positive number, and mostly take the less than mid-way.

So take a startup with $12$ and check the result, then again tried to get zero.

  $\begin{array}{c}{w}^3-2w^2-3w-1056=0\\ {\left(12\right)}^3-2{\left(12\right)}^2-3\left(12\right)-1056=0\\ 1728-2\left(144\right)-36-1056=0\\ 1728-288-36-1056=0\end{array}$

To get zero:

  $\begin{array}{c}1728-288-36-1056=0\\ 348=0\end{array}$

So the value is not zero its too grater than zero.

So check with the value $11$ .

  $\begin{array}{c}{w}^3-2w^2-3w-1056=0\\ {\left(11\right)}^3-2{\left(11\right)}^2-3\left(11\right)-1056=0\\ 1331-2\left(121\right)-33-1056=0\\ 1331-242-36-1056=0\end{array}$

To get zero:

  $\begin{array}{c}1331-242-36-1056=0\\ 0=0\end{array}$

Here got the real root, it means $\left(w-11\right)$ is a factor.

Now divide the expression $w^3-2w^2-3w-1056=0$ to the factor $\left(w-11\right)$ , to check for the other possible solutions.

  $\begin{array}{l}w-11\begin{array}{c}\hfill w^2+9w+96\\ \hfill \overline{)\begin{array}{ccccccc}{w}^3& -& 2w^2& -& 3w& -& 1056\end{array}}\end{array}\\ \begin{array}{ccc}{w}^3& -& 11w^2\end{array}\\ \begin{array}{ccc}9w^2& -& 3w\end{array}\\ \begin{array}{ccc}9w^2& -& 99w\end{array}\\ \begin{array}{ccc}96w& -& 1056\end{array}\\ \begin{array}{ccc}96w& -& 1056\end{array}\\ \begin{array}{c}0\end{array}\end{array}$

So the other factor is:

  $w^2+9w+96$

Now put the value in the formula:

  $b^2-4ac$

Here the values of $a,b,c$ are $a=1,b=9,c=96$

To check the discriminants apply the values in the formula:

  $\begin{array}{c}{b}^2-4ac\\ 9^2-4\left(1\right)\left(96\right)\\ 81-384\\ -303\end{array}$

Its negative so can say that the other two roots are complex.

So, the real root is $w=11$

Since the dimensions are:

  $\begin{array}{c}w=11\\ w+1=12\\ w-3=8\end{array}$

That means:

  $\begin{array}{c}w=11\\ l=12\\ h=8\end{array}$

Thus, the value of length, width and height of a rectangular prism is: $L=12cm,W=11cm,H=8cm$ .