To find the surface area of the triangular prism, we need to calculate the area of the two triangular bases and the three rectangular faces of the prism.
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Area of the triangular base: The triangular base is a right triangle with a height of 1.5 inches and a base (length) of 5 inches.
The area \( A \) of a right triangle is given by: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] Substituting the values: \[ A = \frac{1}{2} \times 5 \text{ in} \times 1.5 \text{ in} = \frac{1}{2} \times 7.5 \text{ in}^2 = 3.75 \text{ in}^2 \] Since there are two triangular bases, the total area of the bases is: \[ 2 \times 3.75 \text{ in}^2 = 7.5 \text{ in}^2 \]
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Areas of the rectangular faces: The prism has three rectangular faces. Their dimensions are:
- The rectangle opposite the base (length = base of triangle = 5 in, height = height of prism = 1.5 in): \[ \text{Area} = 5 \text{ in} \times 1.5 \text{ in} = 7.5 \text{ in}^2 \]
- The rectangle adjacent to the height (length = height of triangle = 1.5 in, height = height of prism = 1.5 in): \[ \text{Area} = 1.5 \text{ in} \times 1.5 \text{ in} = 2.25 \text{ in}^2 \]
- The rectangle adjacent to the hypotenuse (length = hypotenuse = 5.22 in, height = height of prism = 1.5 in): \[ \text{Area} = 5.22 \text{ in} \times 1.5 \text{ in} = 7.83 \text{ in}^2 \]
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Total area of the rectangular faces: Combining the areas of the three rectangles: \[ 7.5 \text{ in}^2 + 2.25 \text{ in}^2 + 7.83 \text{ in}^2 = 17.58 \text{ in}^2 \]
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Total surface area of the prism: The total surface area \( S \) is: \[ S = \text{Area of the bases} + \text{Total area of the rectangles} \] Substituting the values: \[ S = 7.5 \text{ in}^2 + 17.58 \text{ in}^2 = 25.08 \text{ in}^2 \]
Thus, the surface area of the prism is 25.08 in².