To find the surface area of a triangular prism, we can use the following formula:
\[ \text{Surface Area} = \text{Base Area} + \text{Lateral Area} \]
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Calculate the area of the triangular base: The base is a right triangle, so the area \( A \) can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Given:
- Base = 5 inches
- Height = 1.5 inches
Substituting these values:
\[ A = \frac{1}{2} \times 5 \times 1.5 = \frac{1}{2} \times 7.5 = 3.75 \ \text{in}^2 \]
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Calculate the lateral area: The lateral area consists of three rectangular faces. The dimensions of each rectangular face are:
- Rectangle 1 (base of triangle) = height of the prism × base of the triangle
- Rectangle 2 (one leg of triangle) = height of the prism × height of the triangle
- Rectangle 3 (the hypotenuse of the triangle) = height of the prism × hypotenuse of the triangle
Calculating the areas of each rectangle:
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Area of Rectangle 1: \[ A_1 = \text{height of prism} \times \text{base} = 1.5 \times 5 = 7.5 \ \text{in}^2 \]
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Area of Rectangle 2: \[ A_2 = \text{height of prism} \times \text{height of triangle} = 1.5 \times 1.5 = 2.25 \ \text{in}^2 \]
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Area of Rectangle 3: \[ A_3 = \text{height of prism} \times \text{hypotenuse} = 1.5 \times 5.22 = 7.83 \ \text{in}^2 \]
Now, add the areas of the three rectangles:
\[ \text{Lateral Area} = A_1 + A_2 + A_3 = 7.5 + 2.25 + 7.83 = 17.58 \ \text{in}^2 \]
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Total Surface Area: Now, sum the area of the two triangular bases (since there are two of them) and the lateral area:
\[ \text{Surface Area} = 2 \times \text{Base Area} + \text{Lateral Area} = 2 \times 3.75 + 17.58 \]
\[ \text{Surface Area} = 7.5 + 17.58 = 25.08 \ \text{in}^2 \]
Thus, the surface area of the prism is 25.08 in.².
The correct response is:
25.08 in.²