Chapter 8.6, Problem 52PPS

(a)

To determine

To provide: An expression for the height and width of the prism in terms of length.

Explanation of Solution

Given information:

The rectangular prism

  

Proof:

The height of rectangular prism is calculated as when length = 14

Height = 14-8=6.

Hence, height is 6 more than the length.

Thus, height = $=l+6$

The width of rectangular prism is calculated as when length = 14

width = 14-4=10.

Hence, width is 10 more than the length.

Also width = $=l+10$

(b)

To determine

To give: A polynomial for the volume of the prism in terms of length.

Explanation of Solution

Given information:

The rectangular prism

  

Proof:

The volume of a rectangular prism is given as,

  $\text{Volume}=\text{length}\ast {\text{width}}^{\ast }\text{height}$

where V is the volume, length is the l, width is $(l+6)$ and the height is $(l+10)$

Hence,

Using distributive property,

  $\begin{array}{cc}\text{Volume}& =l(l+6)(l+10)\\ & =l^3+16l^2+60l\end{array}$

The volume of a rectangular prism is $l^3+16l^2+60l$