Chapter 1.3, Problem 42E

Solution

To find: the value of $x$ .

The value of $x\text{ is 6}\text{.}$

Given:

A rectangular picnic site measures 24 feet by 10 feet.

Concept Used:

Factoring method to find the roots of the quadratic equation.

Calculation:

Length of rectangle $=24\text{feet}$

Width of rectangle $=10\text{feet}$

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

  $\begin{array}{l}\text{Area of rectangle}=24\times 10\\ =240{\text{ft}}^2\end{array}$

Now, we have to double the area:

New area of rectangle is $=240\times 2{\text{ft}}^2$

  $=480{\text{ft}}^2$

After adding the same distance of $x$ to the length and the width:

New length of rectangle is $\left(x+24\right)\text{ft}$ .

New width of rectangle is $\left(x+10\right)\text{ft}$ .

Here, Applying the area of triangle formula and we get:

  $\therefore \left(x+24\right)\left(x+10\right)=480$

Solving this equation:

  $\begin{array}{c}\left(x+24\right)\left(x+10\right)=480\\ x\left(x+10\right)+24\left(x+10\right)=480\\ x^2+10x+24x+240=480\\ x^2+34x+240-480=0\\ x^2+34x-240=0\end{array}$

Applying the factoring method of quadratic equation:

  $\begin{array}{c}{x}^2+34x-240=0\\ x^2+40x-6x-240=0\\ x\left(x+40\right)-6\left(x+40\right)=0\\ \left(x+40\right)\left(x-6\right)=0\end{array}$

Using Zero Product property:

  $\begin{array}{c}x+40=0\text{or}x-6=0\\ x=-40\text{or}x=6\end{array}$

Here, Neglecting the value of $x=-40$ As distance cannot be negative.

Therefore, The value of $x\text{ is 6}\text{.}$

Conclusion:

The value of $x\text{ is 6}\text{.}$