Chapter 1.5, Problem 6MRPS

Solution

To find: the value of $x$ .

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

Given:

A rectangular vegetable garden measures 42 feet by 8 feet.

Concept Used:

Factoring method to find the roots of the quadratic equation.

Calculation:

Length of rectangle $=42\text{feet}$ .

Width of rectangle $=8\text{feet}$ .

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

  $\begin{array}{l}\text{Area of rectangle}=42\times 8\\ =336{\text{feet}}^2\end{array}$

Now, we have to double the area:

New area of garden is $=336\times 2{\text{feet}}^2$ ,

  $=672{\text{feet}}^2$

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

New length of rectangle is $\left(x+42\right)\text{feet}$ .

New width of rectangle is $\left(x+88\right)\text{feet}$ .

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

  $\therefore \left(x+42\right)\left(x+8\right)=672$

Solving this equation:

  $\begin{array}{c}\left(x+42\right)\left(x+8\right)=672\\ x\left(x+8\right)+42\left(x+8\right)=672\\ x^2+8x+42x+336=672\\ x^2+50x+336-672=0\\ x^2+50x-336=0\end{array}$

Applying the factoring method of quadratic equation:

  $\begin{array}{c}{x}^2+50x-336=0\\ x^2+56x-6x-336=0\\ x\left(x+56\right)-6\left(x+56\right)=0\\ \left(x+56\right)\left(x-6\right)=0\end{array}$

Using Zero Product property:

  $\begin{array}{c}x+56=0\text{or}x-6=0\\ x=-56\text{or}x=6\end{array}$

Here, Neglecting the value of $x=-56$ As distance cannot be negative.

Therefore, the value of $x\text{ is 6 feet}$

Conclusion:

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