Chapter 1.4, Problem 62PS

Solution

To find: the width of the border.

The new dimension of border is 3 feet by 2 feet.

Given:

A rectangular stained glass window measures 2 feet by 1 foot.

Concept Used:

Factoring method to find the roots of the quadratic equation.

Calculation:

Length of glass window $=2\text{feet}$ .

Width of glass window $=1\text{foot}$ .

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

  $\begin{array}{c}\text{Area of glass window}=2\times 1\\ =2{\text{feet}}^2\end{array}$

Now, dimension of glass window with border:

New length of glass window with border is $\left(2x+2\right)\text{feet}$ .

New width of glass window with border is $\left(2x+1\right)\text{feet}$ .

Area of glass is 4 square feet.

Here, Applying the area of rectangle formula:

  $\therefore \left(2x+2\right)\left(2x+1\right)-2\times 1=4$ .

Solving this equation:

  $\begin{array}{c}\left(2x+2\right)\left(2x+1\right)-2\times 1=4\\ \left(2x+2\right)\left(2x+1\right)=4+2\\ 2x\left(2x+1\right)+2\left(2x+1\right)=6\\ 4x^2+2x+4x+2=6\\ 4x^2+6x+2-6=0\\ 4x^2+6x-4=0\\ 2\left(2x^2+3x-2\right)=0\\ \left(2x^2+3x-2\right)=0\\ 2x^2+4x-x-2=0\\ 2x\left(x+2\right)-1\left(x+2\right)=0\\ \left(x+2\right)\left(2x-1\right)=0\end{array}$

Using Zero Product property:

  $\begin{array}{c}x+2=0\text{or}2x-1=0\\ x=-2\text{or}x=\frac{1}{2}\end{array}$

Here, Neglecting the value of $x=-2$ as distance cannot be negative.

Therefore, the new dimension of the border is:

  $\begin{array}{c}\text{Length}=2x+2\\ =2\times \frac{1}{2}+2\\ =3\text{feet}\end{array}$

Also,

  $\begin{array}{c}\text{width}=2x+1\\ =2\times \frac{1}{2}+1\\ =2\text{ feet}\end{array}$

Conclusion:

The new dimension of border is 3 feet by 2 feet.