Question

A swimming pool is to be constructed in a 1,176-ft2 backyard. There is to be a fence that will surround a 14-by-28-foot pool. The pool builder wants to build a concrete paver deck of a uniform width, x, surrounding the pool and filling the entire area of the backyard. What is the width of the pool deck?

7 ft
14 ft
28 ft
56 ft

Answer 1

To determine the width \( x \) of the concrete paver deck surrounding the pool, we can break the problem down into a few steps.

1. Calculate the area of the pool:
\[
\text{Area of the pool} = \text{length} \times \text{width} = 28 \, \text{ft} \times 14 \, \text{ft} = 392 \, \text{ft}^2
\]

2. Find the area available for the deck:
\[
\text{Area of the backyard} = 1,176 \, \text{ft}^2
\]
\[
\text{Area of the deck} = \text{Area of the backyard} - \text{Area of the pool} = 1,176 \, \text{ft}^2 - 392 \, \text{ft}^2 = 784 \, \text{ft}^2
\]

3. Determine the dimensions of the pool plus the deck. The width of the deck on all sides adds \( 2x \) to the dimensions of the pool:
- The total length with the deck: \( 28 + 2x \)
- The total width with the deck: \( 14 + 2x \)

4. Calculate the area of the entire pool and deck:
\[
\text{Area of the entire structure} = (28 + 2x)(14 + 2x)
\]

5. Set up the equation for the area that includes the deck and pool:
\[
(28 + 2x)(14 + 2x) = 1,176
\]

6. Expand and simplify the equation:
\[
28 \times 14 + 56x + 28 \cdot 2x + 14 \cdot 2x + 4x^2 = 1,176
\]
\[
392 + 56x + 28x + 14x + 4x^2 = 1,176
\]
\[
392 + 98x + 4x^2 = 1,176
\]

7. Rearranging gives:
\[
4x^2 + 98x + 392 - 1,176 = 0
\]
\[
4x^2 + 98x - 784 = 0
\]

8. Divide the entire equation by 4 to simplify:
\[
x^2 + 24.5x - 196 = 0
\]

9. Use the quadratic formula to find \( x \):
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-24.5 \pm \sqrt{(24.5)^2 - 4(1)(-196)}}{2(1)}
\]
\[
x = \frac{-24.5 \pm \sqrt{600.25 + 784}}{2}
\]
\[
x = \frac{-24.5 \pm \sqrt{1384.25}}{2}
\]
\[
x = \frac{-24.5 \pm 37.2}{2}
\]

Calculating the two possible results:
1. \( x = \frac{12.7}{2} = 6.35 \) (valid since it's positive)
2. \( x = \frac{-61.7}{2} \) (not valid since it's negative)

Since we need a whole width value, we round \( 6.35 \approx 7 \).

Thus, the width of the pool deck is:
\[
\boxed{7} \text{ ft}
\]