To find the surface area of the triangular prism, we need to calculate two parts: the area of the two triangular bases and the area of the three rectangular faces.
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Finding the area of the triangular base:
The triangular base has side lengths of 6, 5, and 5. We can use Heron's formula to find the area of the triangle.
Calculate the semi-perimeter \( s \):
\[ s = \frac{6 + 5 + 5}{2} = 8 \]
Now, use Heron's formula to find the area \( A \):
\[ A = \sqrt{s(s - a)(s - b)(s - c)} \] where \( a = 6, b = 5, c = 5 \):
\[ A = \sqrt{8(8 - 6)(8 - 5)(8 - 5)} = \sqrt{8 \times 2 \times 3 \times 3} = \sqrt{144} = 12 \]
Therefore, the area of the triangular base is 12.
Since there are two triangular bases, the total area of the two bases is:
\[ 2 \times A = 2 \times 12 = 24 \]
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Finding the area of the rectangular faces:
The rectangular faces have the following dimensions:
- Two rectangles with dimensions \( 6 \) (the base of the triangle) and \( 15 \) (the height of the prism): \[ \text{Area} = 6 \times 15 = 90 \]
- One rectangle with dimensions \( 5 \) (the side of the triangle) and the height of the prism \( 15 \): \[ \text{Area} = 5 \times 15 = 75 \]
- Another rectangle with dimensions \( 5 \) (the other side of the triangle) and the height of the prism \( 15 \): \[ \text{Area} = 5 \times 15 = 75 \]
Now, adding the areas of the three rectangular faces:
\[ \text{Total area of rectangular faces} = 90 + 75 + 75 = 240 \]
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Calculating the surface area of the prism:
Lastly, we sum the areas of the two bases and the three rectangular faces:
\[ \text{Total Surface Area} = \text{Area of bases} + \text{Area of rectangles} \] \[ \text{Total Surface Area} = 24 + 240 = 264 \]
Thus, the total surface area of the triangular prism is:
\[ \boxed{264} \]
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