To find the surface area of a triangular prism, we need to calculate the area of the two triangular bases and the area of the three rectangular lateral faces.
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Area of the triangular face: The triangular face is a right triangle. The area \( A \) of a right triangle can be calculated using the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] Given:
- Base \( = 27.9 , \text{cm} \)
- Height \( = 5.1 , \text{cm} \)
\[ A = \frac{1}{2} \times 27.9 \times 5.1 = 71.145 , \text{cm}^2 \]
Since there are two triangular bases, we multiply by 2: \[ \text{Total area of the triangular bases} = 2 \times 71.145 = 142.29 , \text{cm}^2 \]
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Area of the rectangular faces: The three rectangular faces can be calculated using the lengths of the prism and the sides of the triangle.
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The first rectangle has dimensions:
- Length \( = 30.5 , \text{cm} \)
- Width \( = 27.9 , \text{cm} \)
\[ \text{Area} = 30.5 \times 27.9 = 852.45 , \text{cm}^2 \]
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The second rectangle has dimensions:
- Length \( = 30.5 , \text{cm} \)
- Width \( = 5.1 , \text{cm} \)
\[ \text{Area} = 30.5 \times 5.1 = 155.55 , \text{cm}^2 \]
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The third rectangle has dimensions:
- Length \( = 30.5 , \text{cm} \)
- Width \( = 28.4 , \text{cm} \)
\[ \text{Area} = 30.5 \times 28.4 = 864.2 , \text{cm}^2 \]
Now, adding up the areas of the rectangular faces: \[ \text{Total area of rectangular faces} = 852.45 + 155.55 + 864.2 = 1872.2 , \text{cm}^2 \]
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Calculating the total surface area: Finally, we add the area of the triangular bases to the area of the rectangular faces: \[ \text{Total Surface Area} = 142.29 + 1872.2 = 2014.49 , \text{cm}^2 \]
Rounding to the nearest whole number, the total surface area required for wrapping paper is: \[ \text{S.A.} = 2014 , \text{cm}^2 \]
Therefore, the answer is: \[ \boxed{2014} , \text{cm}^2 \]