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: \( 3x - 5 \) feet
- Width: \( 4x - 10 \) feet
The area \( A_1 \) of the backyard is:
\[ A_1 = \text{Length} \times \text{Width} = (3x - 5)(4x - 10) \]
Let's expand this expression.
\[ (3x - 5)(4x - 10) = 3x \cdot 4x + 3x \cdot (-10) + (-5) \cdot 4x + (-5) \cdot (-10) \]
\[ = 12x^2 - 30x - 20x + 50 \]
\[ = 12x^2 - 50x + 50 \]
### 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:
\[ A_2 = \text{Length} \times \text{Width} = (x + 8)(x - 2) \]
Let's expand this expression.
\[ (x + 8)(x - 2) = x \cdot x + x \cdot (-2) + 8 \cdot x + 8 \cdot (-2) \]
\[ = x^2 - 2x + 8x - 16 \]
\[ = x^2 + 6x - 16 \]
### 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:
\[ A = A_1 - A_2 = (12x^2 - 50x + 50) - (x^2 + 6x - 16) \]
Let's simplify this expression.
\[ A = 12x^2 - 50x + 50 - x^2 - 6x + 16 \]
\[ = 12x^2 - x^2 - 50x - 6x + 50 + 16 \]
\[ = 11x^2 - 56x + 66 \]
So, the area of the region of the yard that they want to cover with new grass is:
\[ 11x^2 - 56x + 66 \text{ ft}^2 \]
Thus, the correct answer is:
\[ 11x^2 - 56x + 66 \text{ ft}^2 \]
So, the correct multiple-choice answer is:
- 11x\(^2\) – 56x + 66 ft\(^2\)
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
1 answer
1 answer