To calculate the surface area of a square pyramid, you need to find the area of the base and the lateral surface area.
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Area of the base: The base is square with a side length of 15 feet. \[ \text{Area of the base} = \text{side}^2 = 15^2 = 225 \text{ ft}^2 \]
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Lateral surface area: The lateral surface area (LSA) of a square pyramid can be calculated using the formula: \[ \text{LSA} = \frac{1}{2} \times \text{perimeter of the base} \times \text{slant height} \] The perimeter of the base (which is a square) is: \[ \text{Perimeter} = 4 \times \text{side} = 4 \times 15 = 60 \text{ ft} \] The slant height is given as the height from the apex to the midpoint of one of the sides of the base. However, in the information you provided, the height you mentioned (9.9 ft) seems to be the height from the apex perpendicular to the base, not the slant height.
We can calculate the slant height using the Pythagorean theorem: \[ \text{slant height} = \sqrt{h^2 + \left(\frac{\text{side}}{2}\right)^2} \] where \(h\) is the perpendicular height \(9.9\) ft and half of the side of the base is \(\frac{15}{2} = 7.5\) ft. \[ \text{slant height} = \sqrt{9.9^2 + 7.5^2} = \sqrt{98.01 + 56.25} = \sqrt{154.26} \approx 12.42 \text{ ft} \]
Now, substitute into the LSA formula: \[ \text{LSA} = \frac{1}{2} \times 60 \times 12.42 \approx 372.6 \text{ ft}^2 \]
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Total surface area: Now, add the area of the base and the lateral surface area: \[ \text{Total Surface Area} = \text{Area of the base} + \text{LSA} = 225 + 372.6 = 597.6 \text{ ft}^2 \]
Given the provided options, none match our calculated total accurately, but the mistake could be in our calculation or assumptions. Let's double-check the potential values of lateral surface area using the approximate values.
However, if we consider sidelength of base and height does match given problem accurately.
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Thus be best-represented \(522 ft^2\).