To find the surface area of the triangular prism, we need to calculate the areas of the two triangular bases and the three rectangular lateral faces.
Step 1: Calculate the area of the triangular base
The triangle has a height of 1.5 inches and a base (length) of 5 inches. The area \(A\) of the triangular base can be calculated using the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] Substituting the values: \[ A = \frac{1}{2} \times 5 \times 1.5 = \frac{7.5}{2} = 3.75 \text{ in}^2 \]
Since there are two triangular bases, the total area of the bases is: \[ \text{Total area of bases} = 2 \times 3.75 = 7.5 \text{ in}^2 \]
Step 2: Calculate the area of the rectangular lateral faces
The prism has three rectangular lateral faces:
- Height \(1.5\) inches and width equal to the length of the base (5 inches).
- Height \(1.5\) inches and width equal to one leg of the triangle.
- Height \(1.5\) inches and width equal to the other leg of the triangle.
To find the lengths of the legs, we can use the Pythagorean theorem for the right triangle:
- Given the base is 5 inches and the height is 1.5 inches, we need to calculate the length of the other leg \(b\).
From the Pythagorean theorem: \[ 5^2 = (1.5)^2 + b^2 \] \[ 25 = 2.25 + b^2 \] \[ b^2 = 25 - 2.25 = 22.75 \] \[ b = \sqrt{22.75} \approx 4.77 \text{ inches} \]
Now we have the dimensions of the legs:
- One leg = 1.5 inches (height of triangle)
- Base = 5 inches
- Other leg = 4.77 inches
Now we can calculate the areas of the rectangular lateral faces:
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Area of rectangle with height 1.5 inches and width 5 inches: \[ A_1 = 1.5 \times 5 = 7.5 \text{ in}^2 \]
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Area of rectangle with height 1.5 inches and width 1.5 inches: \[ A_2 = 1.5 \times 1.5 = 2.25 \text{ in}^2 \]
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Area of rectangle with height 1.5 inches and width 4.77 inches: \[ A_3 = 1.5 \times 4.77 \approx 7.155 \text{ in}^2 \]
Step 3: Calculate the total area of lateral faces
\[ \text{Total area of lateral faces} = A_1 + A_2 + A_3 = 7.5 + 2.25 + 7.155 \approx 16.905 \text{ in}^2 \]
Step 4: Calculate the total surface area of the prism
\[ \text{Total surface area} = \text{Total area of bases} + \text{Total area of lateral faces} \] \[ \text{Total surface area} = 7.5 + 16.905 \approx 24.405 \text{ in}^2 \]
Final Result:
Rounding gives us 24.41 square inches. The closest option to our calculated surface area is:
24.75 in.