Chapter 11.3, Problem 25PPE

To determine

The expression for the missing dimension.

Answer to Problem 25PPE

  $h=4c^2-8c+16$

Explanation of Solution

Given:

The figure,

  

Concept Used:

  $\text{Area of triangle}=\frac{1}{2}\times \left(\text{base}\right)\times \left(\text{height}\right)$

Calculation:

In order to find the expression for the missing dimension in the given triangle, observe that the expression for the area of the triangle and the base length expression is given. And the area of the triangle is obtained by half of the product of the base and height of the triangle. That is,

  $\begin{array}{c}\text{Area of triangle}=\frac{1}{2}\times \left(\text{base}\right)\times \left(\text{height}\right)\\ 2c^3+16=\frac{1}{2}\times \left(c+2\right)\times \left(\text{height}\right)\\ 2\left(2c^3+16\right)=\left(c+2\right)\times \left(\text{height}\right)\\ \frac{4c^3+32}{c+2}=\text{height}\end{array}$

So, in order to find the expression for the height, divide the area by the width expression as shown below

  $c+2\begin{array}{c}\hfill 4c^2-8c+16\\ \hfill \overline{)\begin{array}{l}4c^3+32\\ \underset{\_}{-\left(4c^3+8c^2\right)}\\ -8c^2+32\\ \underset{\_}{-\left(-8c^2-16c\right)}\\ 16c+32\\ \underset{\_}{-\left(16c+32\right)}\\ 0\end{array}}\end{array}$

Thus, the expression for the missing dimension is $h=4c^2-8c+16$ .