Chapter 11.3, Problem 37PPE

To determine

The expression for the height h of the trapezoid.

Answer to Problem 37PPE

  $h=2x+2$

Explanation of Solution

Given:

The area A of a trapezoid is $x^3+2x^2-2x-3$ . The length of its two bases $b_1\text{ and }{b}_2$ are $x\text{ and }{x}^2-3$ .

Concept Used:

  $\text{Area of trapezoid}=\frac{1}{2}\times \left(\text{sum of bases}\right)\times \left(\text{height}\right)$

Calculation:

In order to find the expression for the height h in the given trapezoid, observe that the expression for the area of the trapezoid and the two base length expressions are given. And the area of the trapezoid is obtained by half of the product of the sum of bases and height of the trapezoid. That is,

  $\begin{array}{c}\text{Area of trapezoid}=\frac{1}{2}\times \left(\text{sum of bases}\right)\times \left(\text{height}\right)\\ x^3+2x^2-2x-3=\frac{1}{2}\times \left(x+x^2-3\right)\times \left(h\right)\\ 2\left(x^3+2x^2-2x-3\right)=\left(x+x^2-3\right)\times \left(h\right)\\ \frac{2x^3+4x^2-4x-6}{x+x^2-3}=h\end{array}$

So, in order to find the expression for the height, divide the expressions as shown below

  $x^2+x-3\begin{array}{c}\hfill 2x+2\\ \hfill \overline{)\begin{array}{l}2x^3+4x^2-4x-6\\ \underset{\_}{-\left(2x^3+2x^2-6x\right)}\\ 2x^2+2x-6\\ \underset{\_}{-\left(2x^2+2x-6\right)}\\ 0\end{array}}\end{array}$

Thus, the expression for the height of the trapezoid is $h=2x+2$ .