Chapter 1.3, Problem 59E

Solution

To find: the measure of length and width of rectangle.

Length of rectangle is $9$ units and width is $4$ units.

Given:

Area of rectangle is 36 square units.

Concept Used:

Factoring method to find the roots of the quadratic equation.

Calculation:

From the figure:

Length of rectangle is $(x+5)\text{ unit}$ .

Width of rectangle is $x\text{unit}$ .

Formula of Area of rectangle $=\text{length }\times \text{ width}$ .

  $\therefore \left(x+5\right)x=36$

Solving the above equation:

  $\begin{array}{c}x\left(x+5\right)=36\\ x^2+5x=36\\ x^2+5x-36=0\end{array}$

Applying the factoring method of solving quadratic equation:

  $\begin{array}{c}{x}^2+5x-36=0\\ x^2+9x-4x-36=0\\ x(x+9)-4(x+9)=0\\ (x+9)(x-4)=0\end{array}$

Here, using Zero product property:

  $\begin{array}{c}x+9=0\text{or}x-4=0\\ x=-9\text{or}x=4\end{array}$

Here, Neglecting the value of $x=-9$ , As distance cannot be negative.

  $\begin{array}{l}\therefore \text{ length = 4+5}\\ =9\text{ units}\end{array}$

So,

  $\text{width = 4}\text{units}$

Therefore, length of rectangle is $9$ units and width is $4$ units.

Conclusion:

Length of rectangle is $9$ units and width is $4$ units.