To calculate the surface area of a triangular prism, we need to determine the area of the triangular base and the area of the three rectangular sides.
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Triangular Base Area: The base of the prism is a right triangle given by the dimensions \( W \) (width) and \( H \) (height). The area \( A \) of a triangle is given by: \[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times W \times H = \frac{1}{2} \times 5 \times 2 = 5 , \text{inches}^2 \] Since there are two triangular bases, the total area for the triangular bases is: \[ 2 \times A = 2 \times 5 = 10 , \text{inches}^2 \]
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Rectangular Sides Area: The rectangular sides consist of three rectangles:
- Two rectangles that correspond to the height of the prism and the base of the triangle:
- Rectangle 1 (base W to side length L): \( W \times L = 5 \times 10 = 50 , \text{inches}^2 \)
- Rectangle 2 (base H to side length L): \( H \times L = 2 \times 10 = 20 , \text{inches}^2 \)
- The third rectangle corresponds to the hypotenuse of the triangular base. First, we calculate the hypotenuse using the Pythagorean theorem: \[ \text{Hypotenuse} = \sqrt{W^2 + H^2} = \sqrt{5^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29} \] Then, the area of this rectangle would be: \[ \text{Hypotenuse} \times L = \sqrt{29} \times 10 \approx 5.385 \times 10 = 53.85 , \text{inches}^2 \]
Therefore, the total area for the three rectangular sides is: \[ 50 + 20 + 53.85 \approx 123.85 , \text{inches}^2 \]
- Two rectangles that correspond to the height of the prism and the base of the triangle:
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Total Surface Area (S.A.): Combining the area of the triangular bases and the area of the rectangular sides gives us: \[ S.A. = \text{Area of triangular bases} + \text{Area of rectangular sides} = 10 + 123.85 \approx 133.85 , \text{inches}^2 \]
Given the options you provided, it seems there was some miscalculation or misinterpretation of dimensions, as the closest total of the rectangular sides suggests it should be readjusted.
If we assume that \( S.A. \) corresponds with the rectangular contributions and adjust dimensions accordingly; please review the geometric information you'd provided, and let me know if my interpretation of the dimensions or if the calculations should change.