To find the inequality that can be used to solve for the width [tex]\( x \)[/tex] of the patio, follow these steps:
### 1. Calculate the Area and Cost of the Pond
First, we need to determine the area of the circular pond:
[tex]\[ \text{Area of pond} = \pi \times \text{radius}^2 \][/tex]
Given the radius ([tex]\( r \)[/tex]) of 6 feet:
[tex]\[ \text{Area of pond} = \pi \times 6^2 = 36\pi \][/tex]
### 2. Calculate the Cost of the Pond
Next, find the cost to install the pond. The cost is given as \[tex]$0.62 per square foot:
\[ \text{Cost of pond} = 36\pi \times 0.62 \]
From the result, the cost calculation provides us:
\[ \text{Cost of pond} = 36\pi \times 0.62 = 70.12 \]
### 3. Subtract the Pond Cost from the Total Budget
Stacy's total budget is \$[/tex]536. After paying for the pond, the remaining budget is:
[tex]\[ \text{Remaining budget} = 536 - 70.12 \][/tex]
### 4. Set Up the Inequality for the Patio's Area
The patio's dimensions are given by:
- Width = [tex]\( x \)[/tex]
- Length = [tex]\( x + 13 \)[/tex]
The area of the patio (excluding the pond) is:
[tex]\[ \text{Area of patio} = x \times (x + 13) = x^2 + 13x \][/tex]
The cost to tile this area is \$1 per square foot, therefore the cost is simply the area in square feet.
### 5. Include Remaining Budget into the Equation
Since Stacy's remaining budget after constructing the pond should cover the cost of tiling the patio, we write this as:
[tex]\[ x^2 + 13x \leq 536 - 70.12 \][/tex]
Simplified:
[tex]\[ x^2 + 13x \leq 465.88 \][/tex]
### 6. Form the Final Inequality
We can now include the pond cost directly into the inequality to solve for [tex]\( x \)[/tex]:
[tex]\[ x^2 + 13x \leq 536 - 70.12 \][/tex]
This is equivalent to:
[tex]\[ x^2 + 13x - 70.12 \leq 536 \][/tex]
### Identifying the Correct Option
After simplifying and including the costs properly, the resulting inequality to determine the width [tex]\( x \)[/tex]:
[tex]\[ 1x^2 + 13x - 70.12 \leq 536 \][/tex]
Therefore, the correct answer is given by:
[tex]\[ \boxed{(1 x^2 + 13 x - 70.12 \leq 536)} \][/tex]
In the context of the options provided, this reflected exactly matches the required format for the correct answer.
