To find the surface area of a triangular prism, we need to determine the areas of the triangular bases and the rectangular faces.
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Area of the triangular base: The triangular face is described with side lengths of 10 cm (right side), 10 cm (bottom side), and 8 cm (left side).
To calculate the area of the triangle, we can use Heron's formula: \[ s = \frac{a + b + c}{2} \] where \( a = 10 \), \( b = 10 \), and \( c = 8 \).
First, calculate the semi-perimeter \( s \): \[ s = \frac{10 + 10 + 8}{2} = 14 \]
Now, apply Heron’s formula for the area \( A \): \[ A = \sqrt{s(s - a)(s - b)(s - c)} \] \[ A = \sqrt{14(14 - 10)(14 - 10)(14 - 8)} \] \[ A = \sqrt{14 \times 4 \times 4 \times 6} \] \[ A = \sqrt{14 \times 96} = \sqrt{1344} \] The area of the triangle is approximately: \[ A \approx 36.66 \text{ cm}^2 \]
Since there are two triangular bases: \[ \text{Total area of triangles} = 2 \times 36.66 \approx 73.32 \text{ cm}^2 \]
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Area of the rectangular faces: The prism has three rectangular faces. The dimensions are:
- Two rectangular faces at 10 cm (height of prism) and the length of the prism (denote this length as \( h \), which needs to be assumed or given).
- One rectangular face with base of 8 cm (height) and the length of the prism.
Assuming the length of the prism is given (let's assume it's 10 cm as well), the areas of the rectangular faces would be:
- Two rectangular faces with dimensions \( 10 \text{ cm} \times h \), each contributes: \[ 10 \times 10 = 100 \text{ cm}^2 \text{ each} \]
- One rectangular face with dimensions \( 8 \text{ cm} \times h \): \[ 8 \times 10 = 80 \text{ cm}^2 \]
Thus, the total area for rectangular faces is: \[ 100 + 100 + 80 = 280 \text{ cm}^2 \]
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Total surface area: Combine the areas of the triangular bases and rectangular faces: \[ \text{Total Surface Area} = \text{Area of triangles} + \text{Area of rectangles} \] \[ \text{Total Surface Area} = 73.32 + 280 = 353.32 \text{ cm}^2 \]
Since it appears that we might be misinterpreting the lengths about the triangular face, the choices provided do not match. Please check the lengths and values given/assumed.
Given the options provided, there might be an error in interpretation or lengths. The closest assumption is 280 cm², if we’re only focusing on rectangular calculations combined without reviewing the triangle side area in detail.
The best final choice would be:
- 280 square centimeters.
Would you like me to double-check with different dimensions?