Stacy wants to build a patio with a small, circular pond in her backyard. The pond will have a 6-foot radius. She also wants to install tiles in the remaining area of the patio. The length of the patio is 13 feet longer than the width.

If the cost of installing tiles is [tex]$1 per square foot, and the cost of installing the pond is $[/tex]0.62 per square foot, which of the following inequalities can be used to solve for the width, [tex]\(x\)[/tex], of the patio, if Stacy can spend no more than $536 on this project?

A. [tex]\( x^2 + 13x + 13.68\pi \leq 536 \)[/tex]
B. [tex]\( x^2 + 13x - 13.68\pi \leq 536 \)[/tex]
C. [tex]\( x^2 + 13x - 58.32\pi \geq 536 \)[/tex]
D. [tex]\( 0.62x^2 + 8.06x - 13.68\pi \leq 536 \)[/tex]



Answer :

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]