To determine the width \( x \) of the concrete paver deck surrounding the pool, we can break the problem down into a few steps.
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Calculate the area of the pool: \[ \text{Area of the pool} = \text{length} \times \text{width} = 28 , \text{ft} \times 14 , \text{ft} = 392 , \text{ft}^2 \]
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Find the area available for the deck: \[ \text{Area of the backyard} = 1,176 , \text{ft}^2 \] \[ \text{Area of the deck} = \text{Area of the backyard} - \text{Area of the pool} = 1,176 , \text{ft}^2 - 392 , \text{ft}^2 = 784 , \text{ft}^2 \]
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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: \( 14 + 2x \)
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Calculate the area of the entire pool and deck: \[ \text{Area of the entire structure} = (28 + 2x)(14 + 2x) \]
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Set up the equation for the area that includes the deck and pool: \[ (28 + 2x)(14 + 2x) = 1,176 \]
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Expand and simplify the equation: \[ 28 \times 14 + 56x + 28 \cdot 2x + 14 \cdot 2x + 4x^2 = 1,176 \] \[ 392 + 56x + 28x + 14x + 4x^2 = 1,176 \] \[ 392 + 98x + 4x^2 = 1,176 \]
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Rearranging gives: \[ 4x^2 + 98x + 392 - 1,176 = 0 \] \[ 4x^2 + 98x - 784 = 0 \]
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Divide the entire equation by 4 to simplify: \[ x^2 + 24.5x - 196 = 0 \]
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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:
- \( x = \frac{12.7}{2} = 6.35 \) (valid since it's positive)
- \( x = \frac{-61.7}{2} \) (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: \[ \boxed{7} \text{ ft} \]
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