Chapter 1.3, Problem 67PS

Solution

a.

To find: the area of the existing patio.

  $600{\text{ft}}^2$ .

Given information:

The diagram representing the patio:

  

Formula Used:

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

Calculation:

From the given picture, it is clear that the patio is rectangular-shaped. And the length of the patio is 30 ft, while the width of the patio is 20 ft.

So,

  $\begin{array}{c}\text{Area of the patio}=\text{Length×Width}\\ =30\times 20\\ =600{\text{ft}}^2\end{array}$

b.

To find: a verbal model and an equation to find the value of x .

  $\left(30+x\right)\left(20+x\right)=1064$ .

Given information:

The museum wants to increase the area of the existing patio by 464 square feet by expanding the patio as shown:

  

Formula Used:

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

Calculation:

From part (a), the area of the existing patio is $600{\text{ft}}^2$. The museum wants to increase the area of the existing patio by 464 square feet. So, the new area of the patio is:

  $\begin{array}{c}\text{New Area}=600+464\\ =1064{\text{ft}}^2\end{array}$

Now, by looking the given diagram, it can be seen that the patio is expanded by increasing both its length and the width by x feet. So, the new length of the patio is $30+x$ feet and the new width is $20+x$ feet.

So, the new area is:

  $\begin{array}{c}\text{New Area}=\text{New Length × New Width}\\ =\left(30+x\right)\times \left(20+x\right)\\ =\left(30+x\right)\left(20+x\right)\end{array}$

But it is also known that the new area is $1064{\text{ft}}^2$ . So, the required equation is:

  $\left(30+x\right)\left(20+x\right)=1064$ .

c.

To solve: the equation found in part b and find the value of x .

  $x=8\text{ft}$ .

Given information:

The museum wants to increase the area of the existing patio by 464 square feet by expanding the patio as shown:

  

Property Used:

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

Calculation:

From part (b), the equation to be used to find the value of x is:

  $\left(x+30\right)\left(x+20\right)=1064$ .

Multiply using FOIL,

  $x^2+20x+30x+600=1064$ .

Combine the like terms,

  $x^2+50x+600=1064$ .

Subtract 1064 from both sides,

  $x^2+50x-464=0$ .

Factor the above equation,

  $\begin{array}{c}{x}^2+58x-8x-464=0\\ x\left(x+58\right)-8\left(x+58\right)=0\\ \left(x+58\right)\left(x-8\right)=0\end{array}$

Then, by zero product property,

  $x+58=0$ or $x-8=0$

Solve for x ,

  $x=-58$ or $x=8$

Since the museum wants to increase the patio's width and length by x feet, so the value of x can’t be negative. So, the only solution to the given problem is:

  $x=8\text{ft}$ .

Thus, the length and the width of the patio are increased by 8 feet.