To determine which inequality can be used to solve for the width [tex]\( x \)[/tex] of Stacy’s patio, we will go through the problem step by step.
1. Understanding the dimensions and areas:
- The radius of the pond is [tex]\( 6 \)[/tex] feet.
- The length of the patio is [tex]\( 13 \)[/tex] feet longer than the width [tex]\( x \)[/tex]. Therefore, the length is [tex]\( x + 13 \)[/tex].
2. Calculating the area of the pond:
- The pond is a circle with a radius of [tex]\( 6 \)[/tex] feet.
- The area of the pond is given by the formula for the area of a circle: [tex]\( \text{Area} = \pi r^2 \)[/tex], where [tex]\( r \)[/tex] is the radius.
- Substituting the radius, we get [tex]\( \text{Area}_{pond} = \pi \times 6^2 = 36\pi \)[/tex] square feet.
3. Calculating the area of the patio:
- The patio is rectangular with length [tex]\( x + 13 \)[/tex] and width [tex]\( x \)[/tex].
- The area of the patio is given by the formula for the area of a rectangle: [tex]\( \text{Area}_{patio} = \text{length} \times \text{width} \)[/tex].
- Substituting the given values, we get [tex]\( \text{Area}_{patio} = x \times (x + 13) = x^2 + 13x \)[/tex].
4. Determining the remaining area to be tiled:
- The remaining area to be tiled is the area of the patio minus the area of the pond.
- Therefore, the effective tile area is [tex]\( x^2 + 13x - 36\pi \)[/tex] square feet.
5. Calculating the costs:
- The cost of tiling is given as [tex]$1 per square foot.
- The cost of tiling the remaining area is \( 1 \times (x^2 + 13x - 36\pi) \).
- The cost of building the pond is given as $[/tex]0.62 per square foot.
- The total area of the pond is [tex]\( 36\pi \)[/tex] square feet, hence the cost of the pond is [tex]\( 0.62 \times 36\pi = 22.32\pi \)[/tex].
6. Formulating the cost constraint equation:
- Stacy’s total budget is $536.
- The total expenditure, which includes both tiling and pond construction, should not exceed this budget.
- Therefore, the inequality can be written as:
[tex]\[
1 \times (x^2 + 13x - 36\pi) + 22.32\pi \leq 536
\][/tex]
7. Simplifying the inequality:
- Combine the terms:
[tex]\[
x^2 + 13x - 36\pi + 22.32\pi \leq 536
\][/tex]
- Simplifying further, knowing [tex]\( 36 - 22.32 = 13.68 \)[/tex]:
[tex]\[
x^2 + 13x + 13.68\pi \leq 536
\][/tex]
Therefore, the correct inequality that can be used to solve for the width [tex]\( x \)[/tex] of the patio is:
A. [tex]\( 1 x^2 + 13 x + 13.68 \pi \leq 536 \)[/tex]
