To find the surface area of the square pyramid, we will calculate the area of the square base and the area of the four triangular faces.
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Area of the square base: \[ \text{Area}_{\text{base}} = \text{side}^2 = 15^2 = 225 , \text{ft}^2 \]
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Area of one triangular face: The area of a triangle is given by the formula: \[ \text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} \] In this case, the base of each triangle is the side of the square base (15 feet), and the height is the slant height. To find the slant height, we can use the Pythagorean theorem.
The height from the apex of the pyramid to the center of the base creates a right triangle with:
- One leg being the height of the pyramid (9.9 feet)
- The other leg being half the side of the square base (7.5 feet)
Using the Pythagorean theorem: \[ \text{slant height}^2 = 9.9^2 + 7.5^2 \] \[ \text{slant height} = \sqrt{9.9^2 + 7.5^2} = \sqrt{98.01 + 56.25} = \sqrt{154.26} \approx 12.43 , \text{ft} \]
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Area of one triangular face with slant height: \[ \text{Area}_{\text{triangle}} = \frac{1}{2} \times 15 \times 12.43 \approx \frac{1}{2} \times 15 \times 12.43 \approx 93.225 , \text{ft}^2 \]
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Total area of the four triangular faces: \[ \text{Total area}_{\text{triangles}} = 4 \times 93.225 \approx 372.9 , \text{ft}^2 \]
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Total surface area of the pyramid: \[ \text{Total surface area} = \text{Area}{\text{base}} + \text{Total area}{\text{triangles}} = 225 + 372.9 = 597.9 , \text{ft}^2 \]
Since the options do not include 597.9 ft², it seems the correct computation was not aligned with the choices provided, however based on calculations alone:
The total surface area is approximately \(597.9 , \text{ft}^2\).
None of the provided options of 522 ft², 148.5 ft², 819 ft², or 299.25 ft² match our calculation. Therefore, it seems that there may be a misinterpretation of the question or the image may have needed to be checked for values before conclusions could be drawn.