Chapter 1.4, Problem 51E

Solution

To find: The value of $x$ when the area of the rectangle is 30 and, length and width of the rectangle are given to be $3x+1$ and $x$ respectively.

  $x=3$

Given information:

The area of the rectangle is 30 and, length and width of the rectangle are given to be $3x+1$ and $x$ respectively.

Concept used:

Formula for the area of a rectangle is given as $\text{Area}=\text{length}\times \text{ width}$ .

Calculation:

Substitute $x$ for width and $3x+1$ for the length of the rectangle and 30 for the area in the formula for the area of the rectangle, and solve the resulting equation.

  $\begin{array}{c}\text{Area}=\text{length}\times \text{ width}\\ 30=\left(3x+1\right)x\\ 30=3x^2+x\\ 3x^2+x=30\\ 3x^2+x-30=30-30\\ 3x^2+x-30=0\\ \left(3x^2-9x\right)+\left(10x-30\right)=0\\ 3x\left(x-3\right)+10\left(x-3\right)=0\\ \left(x-3\right)\left(3x+10\right)=0\\ \begin{array}{ccc}x-3=0& \text{or}& 3x+10=0\\ x=3& \text{or}& x=-\frac{10}{3}\end{array}\end{array}$

Since the length cannot be negative, the value $x=-\frac{10}{3}$ is discarded. So, the value of $x$ is $3$ .

Conclusion:

The value of $x$ is $3$ .