Chapter 1.4, Problem 52E

Solution

To find: The value of $x$ when the area of the triangle is 115 and, base and height of the triangle are given to be $5x-2$ and $2x$ respectively.

  $x=5$

Given information:

The area of the rectangle is 115 and, length and width of the rectangle are given to be $5x-2$ and $2x$ respectively.

Concept used:

Formula for the area of a triangle is given as $\text{Area}=\frac{1}{2}\times \text{base}\times \text{ height}$ .

Calculation:

Substitute $2x$ for height and $5x-2$ for the base of the triangle and 115 for the area in the formula for the area of the triangle, and solve the resulting equation.

  $\begin{array}{c}\text{Area}=\frac{1}{2}\times \text{base}\times \text{ height}\\ 115=\frac{1}{2}\times \left(5x-2\right)\times 2x\\ 115\cdot 2=\frac{1}{2}\left(5x-2\right)\cdot 2x\cdot 2\\ 230=2x\left(5x-2\right)\\ 230=10x^2-4x\\ 10x^2-4x=230\\ 10x^2-4x-230=230-230\\ 10x^2-4x-230=0\\ 2\left(5x^2-2x-115\right)=0\\ 2\left(5x+23\right)\left(x-5\right)=0\\ \left(5x+23\right)\left(x-5\right)=0\\ \begin{array}{ccc}5x+23=0& \text{or}& x-5=0\\ x=-\frac{23}{5}& \text{or}& x=5\end{array}\end{array}$

Since the length cannot be negative, the value $x=-\frac{23}{5}$ is discarded. So, the value of $x$ is 5.

Conclusion:

The value of $x$ is 5.