Chapter 1.3, Problem 65PS

Solution

To find: the new dimension of rectangular skate park.

The new dimension of skate park is $150\text{ feet}$ long and $\text{100}\text{feet}$ wide.

Given:

A rectangular skate park measures 100 feet long by 50 feet wide.

Concept Used:

Factoring method to find the roots of the quadratic equation.

Calculation:

Length of rectangle $=100\text{feet}$ .

Width of rectangle $=50\text{feet}$ .

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

  $\begin{array}{l}\text{Area of rectangle}=100\times 50\\ =5000{\text{feet}}^2\end{array}$

Now, city wants to triple the area:

New area of skate park is $=5000\times 3{\text{feet}}^2$ .

  $=15000{\text{feet}}^2$ .

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

New length of skate park is $\left(x+100\right)\text{feet}$ .

New width of skate park is $\left(x+50\right)\text{feet}$ .

Here, Applying the area of rectangle formula:

  $\therefore \left(x+100\right)\left(x+50\right)=15000$ .

Solving this equation:

  $\begin{array}{c}\left(x+100\right)\left(x+50\right)=15000\\ x\left(x+50\right)+100\left(x+50\right)=15000\\ x^2+50x+100x+5000=15000\\ x^2+150x+5000-15000=0\\ x^2+150x-10000=0\end{array}$

Applying the factoring method of quadratic equation:

  $\begin{array}{c}{x}^2+150x-10000=0\\ x^2+200x-50x-10000=0\\ x\left(x+200\right)-50\left(x+200\right)=0\\ \left(x+200\right)\left(x-50\right)=0\end{array}$

Using Zero Product property:

  $\begin{array}{c}x+200=0\text{or}x-50=0\\ x=-200\text{or}x=50\end{array}$

Here, Neglecting the value of $x=-200$ As distance cannot be negative.

Therefore, the new dimension of the skate park is:

  $\begin{array}{c}\text{Length}=x+100\\ =50+100\\ =150\text{feet}\end{array}$

Also,

  $\begin{array}{c}\text{width}=x+50\\ =50+50\\ =100\text{ feet}\end{array}$

Conclusion:

The new dimension of skate park is $150\text{ feet}$ long and $\text{100}\text{feet}$ wide.