To find the dimensions of the pool, let's define some variables.
Let:
- \( L \) = length of the pool
- \( W \) = width of the pool
According to the problem, we have the following relationships:
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The width of the pool is 23 feet less than the length of the pool: \[ W = L - 23 \]
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The total area enclosed by the outer edge of the walkway is 1674 square feet. The walkway is 2 feet wide, which means that the dimensions of the area including the walkway will be:
- Length including walkway: \( L + 4 \) (2 ft on each side)
- Width including walkway: \( W + 4 \) (2 ft on each side)
Thus, the area including the walkway can be expressed as: \[ \text{Area} = (L + 4)(W + 4) \] We know this area equals 1674 square feet: \[ (L + 4)(W + 4) = 1674 \]
Now we can substitute \( W \) from the first equation into this equation: \[ (L + 4)((L - 23) + 4) = 1674 \] This simplifies to: \[ (L + 4)(L - 19) = 1674 \]
Next, we will expand and simplify this equation: \[ L^2 - 19L + 4L - 76 = 1674 \] \[ L^2 - 15L - 76 = 1674 \] Now, bring 1674 to the left side: \[ L^2 - 15L - 1750 = 0 \]
Now, we will solve this quadratic equation using the quadratic formula, \( L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -15 \), and \( c = -1750 \): \[ L = \frac{-(-15) \pm \sqrt{(-15)^2 - 4 \cdot 1 \cdot (-1750)}}{2 \cdot 1} \] \[ L = \frac{15 \pm \sqrt{225 + 7000}}{2} \] \[ L = \frac{15 \pm \sqrt{7225}}{2} \] Calculating \( \sqrt{7225} = 85 \): \[ L = \frac{15 \pm 85}{2} \]
Calculating the two potential solutions for \( L \):
- \( L = \frac{100}{2} = 50 \)
- \( L = \frac{-70}{2} = -35 \) (not physically meaningful)
Thus the valid length of the pool is \( L = 50 \) feet.
Now we can find the width of the pool: \[ W = L - 23 = 50 - 23 = 27 \text{ feet} \]
Therefore, the dimensions of the pool are:
- Length: \( 50 \) feet
- Width: \( 27 \) feet
To confirm, let's check the total area including the walkway:
- Length including walkway: \( 50 + 4 = 54 \) feet
- Width including walkway: \( 27 + 4 = 31 \) feet
- Area: \( 54 \times 31 = 1674 \) square feet, which matches.
Thus, the dimensions of the pool are: \[ \boxed{50 \text{ feet (length)} \text{ and } 27 \text{ feet (width)}} \]