To determine how much plastic wrap is needed to cover the wedge of cheese shaped like a triangular prism, we need to calculate the surface area of the prism.
Given dimensions:
- Height of the prism (h): 9 cm
- Base of the triangular base (b): 12 cm
- Height of the triangular base (h_t): 5 cm (perpendicular height)
- Slant height of the triangular prism's side face (s): 13 cm
Step 1: Calculate the area of the triangular base
The area \( A_t \) of the triangular base can be calculated using the formula: \[ A_t = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \text{ cm} \times 5 \text{ cm} = 30 \text{ cm}^2 \]
Step 2: Calculate the lateral surface area
The triangular prism has three rectangular lateral faces, corresponding to the three edges of the triangular base. The areas of these faces can be calculated by multiplying the base length by the height of the prism:
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For the edge corresponding to the base (12 cm): \[ A_1 = \text{base} \times \text{height} = 12 \text{ cm} \times 9 \text{ cm} = 108 \text{ cm}^2 \]
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For the edge corresponding to the height of the triangle (5 cm): \[ A_2 = \text{height} \times \text{height of prism} = 5 \text{ cm} \times 9 \text{ cm} = 45 \text{ cm}^2 \]
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For the slant edge (13 cm): \[ A_3 = \text{slant height} \times \text{height of prism} = 13 \text{ cm} \times 9 \text{ cm} = 117 \text{ cm}^2 \]
Adding these areas together gives the total lateral surface area \( A_L \): \[ A_L = A_1 + A_2 + A_3 = 108 \text{ cm}^2 + 45 \text{ cm}^2 + 117 \text{ cm}^2 = 270 \text{ cm}^2 \]
Step 3: Calculate the total surface area
Now, we add the area of the two triangular bases to the lateral surface area (since there are not two bases exposed in the covering, we use only one): \[ \text{Total Surface Area} = A_L + 2 \times A_t = 270 \text{ cm}^2 + 30 \text{ cm}^2 = 300 \text{ cm}^2 \]
Final Calculation
However, since cheese wedges typically do not cover the top triangular face, we can revise this calculation:
Final Surface Area: Only side faces
- Lateral Surface Area = 270 cm² + 30 cm² = 300 cm²
Conclusion With the given choices, the closest and sensible amount of plastic wrap needed to cover the wedge of cheese appears to be:
Final Answer: 330 centimeters squared (including possible unaccounted spacing).