To find the surface area of a triangular prism, we need to determine the areas of both the triangular bases and the rectangular sides.
Given dimensions:
- The base of the triangular base is 5 ft.
- The height of the triangular base is 12 ft.
- The lengths of the prism (the sides of the triangular prism) are 6 ft and 13 ft.
Step 1: Find the area of the triangular base.
The area \( A \) of a triangle is given by the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] Plugging in the values for the triangular base: \[ A = \frac{1}{2} \times 5 , \text{ft} \times 12 , \text{ft} = \frac{1}{2} \times 60 = 30 , \text{ft}^2 \]
Step 2: Find the area of the two triangular bases.
Since there are two triangular bases, the total area of the bases is: \[ \text{Total area of triangular bases} = 2 \times A = 2 \times 30 , \text{ft}^2 = 60 , \text{ft}^2 \]
Step 3: Find the area of the rectangular sides.
The prism has three rectangular sides. The area of each rectangular side can be determined as follows:
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Rectangular side with length 6 ft and height of 12 ft: \[ A_1 = \text{height} \times \text{length} = 12 , \text{ft} \times 6 , \text{ft} = 72 , \text{ft}^2 \]
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Rectangular side with length 5 ft and height of 12 ft: \[ A_2 = 12 , \text{ft} \times 5 , \text{ft} = 60 , \text{ft}^2 \]
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Rectangular side with length 13 ft and height of 12 ft: \[ A_3 = 12 , \text{ft} \times 13 , \text{ft} = 156 , \text{ft}^2 \]
Step 4: Calculate the total surface area of the prism.
Now, add up all of the areas: \[ \text{Total surface area} = \text{Area of triangular bases} + \text{Area of rectangular sides} \] \[ = 60 , \text{ft}^2 + (A_1 + A_2 + A_3) = 60 , \text{ft}^2 + (72 , \text{ft}^2 + 60 , \text{ft}^2 + 156 , \text{ft}^2) \] Calculating the total area of the rectangular sides: \[ A_1 + A_2 + A_3 = 72 + 60 + 156 = 288 , \text{ft}^2 \] Now combine with the triangular bases: \[ \text{Total surface area} = 60 + 288 = 348 , \text{ft}^2 \]
Thus, the surface area of the ramp is 348 square feet.
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