Chapter 1.4, Problem 63PS

Solution

To find: the width of the border of yellow roses.

  $x=2\text{ft}$ .

Given information:

The inside rectangular park contains the red roses flower bed and the outer border contains the yellow roses. There are same numbers of red and yellow roses, so the area of the border and the flower bed will be equal. The diagram given below depicts the given scenario:

  

Formula Used:

  $\text{Area of rectangle}=\text{Length}\times \text{Width}$

Property Used:

Zero Product property: If product of two expressions is zero, then either one or both the expressions equal zero.

Calculation:

From the given picture, it is clear that the flower bed (containing red roses) is rectangular-shaped. And its length of is 12 ft, while its width is 8 ft.

So,

  $\begin{array}{c}\text{Area of the flower bed}=\text{Length×Width}\\ =12\times 8\\ =96f{t}^2\end{array}$

Now, by looking the given diagram, it can be seen that the total length of the park (including flower bed and the border) is $12+2x$ feet and the total width is $8+2x$ feet.

So, the total area of the park is:

  $\begin{array}{c}\text{Total Area}=\text{Total Length × Total Width}\\ =\left(12+2x\right)\times \left(8+2x\right)\\ =\left(2x+12\right)\left(2x+8\right)\\ =4\left(x+6\right)\left(x+4\right)\end{array}$

Now it is also known that the area of flower bed is equal to the area of the border. So, the total area of the park is basically two times the area of the flower bed.

  $\begin{array}{c}\text{Total Area}=2\times 96\\ =192{\text{ft}}^2\end{array}$

So, this gives the equation

  $4\left(x+6\right)\left(x+4\right)=192$

Divide both sides by 4,

  $\left(x+6\right)\left(x+4\right)=48$

Multiply using FOIL,

  $x^2+4x+6x+24=48$

Combine the like terms,

  $x^2+10x+24=48$

Subtract 48 from both sides,

  $x^2+10x-24=0$

Factor the above equation,

  $\begin{array}{c}{x}^2+12x-2x-24=0\\ x\left(x+12\right)-2\left(x+12\right)=0\\ \left(x+12\right)\left(x-2\right)=0\end{array}$

Then, by zero product property,

  $x+12=0$ or $x-2=0$

Solve for x ,

  $x=-12$ or $x=2$

Since here x represents distance, so the value of x can’t be negative. So, the only solution to the given problem is:

  $x=2\text{ft}$ .

Thus, the width of the border of yellow roses is 2 feet.