To determine the amount of plastic wrap needed to cover the wedge of cheese shaped like a triangular prism, we need to calculate the surface area of the prism. A triangular prism has two triangular bases and three rectangular lateral faces.
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Area of the triangular base: The triangular base is a right triangle, and its area can be calculated using the formula: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] Here, the base is 9 cm and the height (perpendicular side) is 5 cm: \[ \text{Area of triangle} = \frac{1}{2} \times 9 \times 5 = \frac{1}{2} \times 45 = 22.5 , \text{cm}^2 \]
Since there are two triangular bases: \[ \text{Total area of triangular bases} = 2 \times 22.5 = 45 , \text{cm}^2 \]
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Area of the rectangular lateral faces: There are three rectangular lateral faces. Their areas depend on the dimensions of the prism:
- One rectangle using the hypotenuse (13 cm): \[ \text{Area} = \text{hypotenuse} \times \text{length} = 13 \times 12 = 156 , \text{cm}^2 \]
- One rectangle using the base (9 cm): \[ \text{Area} = 9 \times 12 = 108 , \text{cm}^2 \]
- One rectangle using the height (5 cm): \[ \text{Area} = 5 \times 12 = 60 , \text{cm}^2 \]
Now, sum up the areas of the three rectangular faces: \[ \text{Total area of rectangular lateral faces} = 156 + 108 + 60 = 324 , \text{cm}^2 \]
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Total Surface Area: Finally, add the areas of the triangular bases and the lateral faces together: \[ \text{Total Surface Area} = \text{Area of bases} + \text{Area of lateral faces} = 45 + 324 = 369 , \text{cm}^2 \]
Thus, the amount of plastic wrap needed to cover the wedge of cheese is 369 cm².
Since the options provided (287 cm², 330 cm², 810 cm², 240 cm²) do not include 369 cm², if the question and values are correct, there may be a discrepancy or missing information in the options. Please check the dimensions and the problem details again for accuracy.