To find the surface area of the triangular prism, we need to calculate the surface area of the triangular bases and the rectangular sides.
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Area of the triangular base: The triangular base is an equilateral triangle with each side measuring 10 cm. The formula for the area of an equilateral triangle is given by: \[ A = \frac{\sqrt{3}}{4} s^2 \] Where \( s \) is the length of a side.
For our triangle: \[ A = \frac{\sqrt{3}}{4} (10)^2 = \frac{\sqrt{3}}{4} \cdot 100 = 25\sqrt{3} , \text{square centimeters} \]
Since there are two triangular bases: \[ \text{Total area of two triangular bases} = 2 \cdot 25\sqrt{3} = 50\sqrt{3} , \text{square centimeters} \]
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Area of the rectangular faces: The prism has three rectangular faces:
- Two rectangles measuring \( 10 , \text{cm} \times 6 , \text{cm} \)
- One rectangle measuring \( 10 , \text{cm} \times 8 , \text{cm} \) (the height of the triangle multiplied by the prism’s height)
The area of each rectangular face:
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Area of the two \( 10 , \text{cm} \times 6 , \text{cm} \) faces: \[ 2 \cdot (10 \cdot 6) = 2 \cdot 60 = 120 , \text{square centimeters} \]
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Area of the \( 10 , \text{cm} \times 8 , \text{cm} \) face: \[ 10 \cdot 8 = 80 , \text{square centimeters} \]
So, total area of the rectangular faces: \[ 120 + 80 = 200 , \text{square centimeters} \]
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Total surface area: Now, we can calculate the total surface area of the prism: \[ \text{Total surface area} = \text{Area of triangular bases} + \text{Area of rectangular faces} \]
Given \( \sqrt{3} \approx 1.732 \): \[ 50\sqrt{3} \approx 50 \cdot 1.732 \approx 86.6 , \text{square centimeters} \] Therefore, the approximate total surface area: \[ \text{Total surface area} \approx 86.6 + 200 \approx 286.6 , \text{square centimeters} \]
However, noticing that the choices provided do not include this value suggests that the area of the triangle previously calculated should be re-evaluated.
Let's use the given height of the triangle which is 8 cm: The area of the triangle can also be calculated as: \[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \times 8 = 40 , \text{square centimeters} \]
So the total area of two triangular bases becomes: \[ \text{Total area of two triangular bases} = 2 \cdot 40 = 80 , \text{square centimeters} \]
Putting that back into our total surface area: \[ \text{Total surface area} = 80 + 200 = 280 , \text{square centimeters} \]
Thus, the correct response for the surface area of the triangular prism is: 280 square centimeters.