Chapter 1.3, Problem 66PS

Solution

To find: the new dimension of rectangular enclosure of zoo.

The new dimension of enclosure is $45\text{ feet}$ long and $\text{28 feet}$ wide.

Given:

A rectangular enclosure of zoo measures 35 feet by 18 feet.

Concept Used:

Factoring method to find the roots of the quadratic equation.

Calculation:

Length of rectangle $=35\text{feet}$

Width of rectangle $=18\text{feet}$

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

  $\begin{array}{l}\text{Area of rectangle}=35\times 18\\ =630{\text{feet}}^2\end{array}$

Now, we have to double the area:

New area of rectangle is $=630\times 2{\text{feet}}^2$

  $=1260{\text{feet}}^2$

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

New length of rectangle is $\left(x+35\right)\text{feet}$ .

New width of rectangle is $\left(x+18\right)\text{feet}$ .

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

  $\therefore \left(x+35\right)\left(x+18\right)=1260$

Solving this equation:

  $\begin{array}{c}\left(x+35\right)\left(x+18\right)=1260\\ x\left(x+18\right)+35\left(x+18\right)=1260\\ x^2+18x+35x+630=1260\\ x^2+53x+630-1260=0\\ x^2+53x-630=0\end{array}$

Applying the factoring method of quadratic equation:

  $\begin{array}{c}{x}^2+53x-630=0\\ x^2+63x-10x-630=0\\ x\left(x+63\right)-10\left(x+63\right)=0\\ \left(x+63\right)\left(x-10\right)=0\end{array}$

Using Zero Product property:

  $\begin{array}{c}x+63=0\text{or}x-10=0\\ x=-63\text{or}x=10\end{array}$

Here, Neglecting the value of $x=-63$ As distance cannot be negative.

Therefore, the new dimension of the enclosure is:

  $\begin{array}{c}\text{Length}=x+35\\ =10+35\\ =45\text{feet}\end{array}$

Also,

  $\begin{array}{c}\text{width}=x+18\\ =10+18\\ =28\text{ feet}\end{array}$

Conclusion:

The new dimension of enclosure is $45\text{ feet}$ long and $\text{28 feet}$ wide.