12.

Question

12. 
The Johnsons want to cover their backyard with new grass. Their backyard is rectangular, with a length of 3x – 5 feet and a width of 4x – 10 feet. However, their rectangular swimming pool, along with its surrounding patio, has dimensions of x + 8 by x – 2 feet. What is the area of the region of the yard that they want to cover with new grass?

11x2 – 56x + 66 ft2

12x2 – 50x + 50 ft2

x2 + 6x – 16 ft2

6x2 – 55x + 104 ft2

GPT-4o · Answer

To determine the area of the region of the yard that the Johnsons want to cover with new grass, we need to follow these steps:

1. Calculate the area of the backyard.
2. Calculate the area of the swimming pool and its surrounding patio.
3. Subtract the area of the swimming pool and patio from the area of the backyard.

Let's solve this step-by-step.

### Step 1: Calculate the Area of the Backyard

The dimensions of the backyard are:
- Length:

3 x - 5

feet
- Width:

4 x - 10

feet

The area

A_{1}

of the backyard is:

A_{1} = Length \times Width = (3 x - 5) (4 x - 10)

Let's expand this expression.

(3 x - 5) (4 x - 10) = 3 x \cdot 4 x + 3 x \cdot (- 10) + (- 5) \cdot 4 x + (- 5) \cdot (- 10)

= 12 x^{2} - 30 x - 20 x + 50

= 12 x^{2} - 50 x + 50

### Step 2: Calculate the Area of the Swimming Pool and Patio

The dimensions of the swimming pool and patio are:
- Length:

x + 8

feet
- Width:

x - 2

feet

The area

A_{2}

of the swimming pool and patio is:

A_{2} = Length \times Width = (x + 8) (x - 2)

Let's expand this expression.

(x + 8) (x - 2) = x \cdot x + x \cdot (- 2) + 8 \cdot x + 8 \cdot (- 2)

= x^{2} - 2 x + 8 x - 16

= x^{2} + 6 x - 16

### Step 3: Calculate the Area to be Covered with New Grass

The area to be covered with new grass

A

is the area of the backyard minus the area of the swimming pool and patio:

A = A_{1} - A_{2} = (12 x^{2} - 50 x + 50) - (x^{2} + 6 x - 16)

Let's simplify this expression.

A = 12 x^{2} - 50 x + 50 - x^{2} - 6 x + 16

= 12 x^{2} - x^{2} - 50 x - 6 x + 50 + 16

= 11 x^{2} - 56 x + 66

So, the area of the region of the yard that they want to cover with new grass is:

11 x^{2} - 56 x + 66 {ft}^{2}

Thus, the correct answer is:

11 x^{2} - 56 x + 66 {ft}^{2}

So, the correct multiple-choice answer is:
- 11x

^{2}

– 56x + 66 ft

^{2}