Chapter 6.2, Problem 64SR

To determine

To find: The dimension of the trapezoid area of $4104$ square feet.

Answer to Problem 64SR

The dimension is $25$ , $83$ and $76$ .

Explanation of Solution

Given:

The longer base is 8 more than 3 times of the shorter base and height is 1 more than 3 times of the shorter base.

Calculation:

Let the shorter base be $x$ . Then

  $\begin{array}{c}\text{Longer}\text{base}=8+3x\dots \left(1\right)\\ \text{Height}=1+3x\dots \left(2\right)\end{array}$

The area of the trapezoid is given by $A=\left(\frac{lb+sb}{2}\right)h$ .

Substitute all the values in the area formula.

  $\begin{array}{c}4104=\left(\frac{8+3x+x}{2}\right)\left(1+3x\right)\\ =\left(\frac{8+4x}{2}\right)\left(1+3x\right)\\ =\frac{2\left(4+2x\right)\left(1+3x\right)}{2}\\ =\left(4+2x\right)\left(1+3x\right)\end{array}$

Simplify further.

  $\begin{array}{c}4+12x+2x+6x^2=4104\\ 6x^2+14x-4100=0\\ x^2+\frac{14}{6}x-\frac{4100}{6}=0\\ x^2+\frac{7}{3}x-\frac{2050}{3}=0\end{array}$

Factorize the above equation.

  $\begin{array}{c}{x}^2-25x+\frac{82}{3}x-\frac{2050}{3}=0\\ x\left(x-25\right)+\frac{82}{3}\left(x-25\right)=0\\ \left(x-25\right)\left(x+\frac{82}{3}\right)=0\end{array}$

Simplify further.

  $\begin{array}{c}x-25=0\\ x=25\\ x+\frac{82}{3}=0\\ x=-\frac{82}{3}\end{array}$

Take $x=25$ .

Substitute $x=25$ in the equation (1).

  $\begin{array}{c}\text{Longer}\text{base}=8+3\left(25\right)\\ =8+75\\ =83\end{array}$

Substitute $x=25$ in the equation (2).

  $\begin{array}{c}\text{Height}=1+3\left(25\right)\\ =1+75\\ =76\end{array}$

Therefore, the dimension is $25$ , $83$ and $76$ .