Chapter 11.3, Problem 49PPE

To determine

The area of the shaded base of the prism.

Answer to Problem 49PPE

  $m^2+5m+4$

Explanation of Solution

Given:

The volume of the rectangular prism shown below is $m^3+8m^2+19m+12$ .

  

Concept Used:

  $\text{Volume of rectangular prism}=\left(\text{Area of base}\right)\times \left(\text{height}\right)$

Calculation:

In order to find the expression for the area of the base of the prism, observe that the height of the prism is m +3, and volume is $m^3+8m^2+19m+12$ . So, on using the above formula, it gives

  $\begin{array}{c}\text{Volume of rectangular prism}=\left(\text{Area of base}\right)\times \left(\text{height}\right)\\ m^3+8m^2+19m+12=\left(\text{Area of base}\right)\times \left(m+3\right)\\ \frac{m^3+8m^2+19m+12}{m+3}=\text{Area of base}\end{array}$

So, in order to find the expression for the area if the base, divide the expressions as shown below

  $m+3\begin{array}{c}\hfill m^2+5m+4\\ \hfill \overline{)\begin{array}{l}{m}^3+8m^2+19m+12\\ \underset{\_}{-\left(m^3+3m^2\right)}\\ 5m^2+19m\\ \underset{\_}{-\left(5m^2+15m\right)}\\ 4m+12\\ \underset{\_}{-\left(4m+12\right)}\\ 0\end{array}}\end{array}$

Thus, the area of the base is $m^2+5m+4$ .