To determine how much plastic wrap is needed to cover the triangular prism (the wedge of cheese), we need to calculate the surface area of the prism.
The surface area \( S \) of a triangular prism can be calculated using the formula:
\[ S = (Base , Area) + (Perimeter , of , Base \times Height) \]
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Find the area of the triangular face: The triangular face is a right triangle with a base of 9 cm and a height of 5 cm. The area \( A \) of a triangle is given by:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 9 \times 5 = \frac{45}{2} = 22.5 , \text{cm}^2 \]
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Calculate the base perimeter: The perimeter \( P \) of the triangle can be calculated by adding the lengths of its sides. The edges of the right triangle are 9 cm (base), 5 cm (perpendicular height), and 13 cm (hypotenuse):
\[ P = 9 + 5 + 13 = 27 , \text{cm} \]
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Calculate the lateral area: The height (length of the prism) is 12 cm. The lateral surface area (the area of the three rectangles connecting the triangular bases) can be calculated as:
\[ Lateral , Area = P \times \text{Height} = 27 \times 12 = 324 , \text{cm}^2 \]
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Total surface area: The total surface area is the lateral area plus the areas of the two triangular bases:
\[ S = 2 \times A + Lateral , Area = 2 \times 22.5 + 324 = 45 + 324 = 369 , \text{cm}^2 \]
Given this calculation, none of the options provided (287 cm², 330 cm², 810 cm², 240 cm²) aligns with our derived total surface area of 369 cm². Could you please double-check the dimensions and the options given?