To find the surface area of a prism with a trapezoidal base, we need to calculate the area of the trapezoidal base and then add the area of the rectangular lateral faces of the prism.
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Area of the base (trapezoid): The area \( A \) of a trapezoid is given by the formula:
\[ A = \frac{1}{2} \times (b_1 + b_2) \times h \]
where \( b_1 \) and \( b_2 \) are the lengths of the two bases, and \( h \) is the height of the trapezoid.
Here:
- \( b_1 = 4 , \text{inches} \)
- \( b_2 = 10 , \text{inches} \)
- \( h = 4 , \text{inches} \)
Plugging in the values:
\[ A = \frac{1}{2} \times (4 + 10) \times 4 = \frac{1}{2} \times 14 \times 4 = 28 , \text{in}^2 \]
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Area of the two trapezoidal bases: Since the prism has two bases, we multiply the area of one base by 2:
\[ \text{Total area of the bases} = 28 , \text{in}^2 \times 2 = 56 , \text{in}^2 \]
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Calculating the lateral surface area: The prism has 4 lateral faces that are rectangles. The height of the prism is 10 inches.
To find the lateral surface area, we need to find the lengths of the non-parallel sides of the trapezoid, which we can find using the Pythagorean theorem. The distance between the two bases is the difference in their lengths divided by 2, which gives:
\[ \text{Length of left and right segments} = \frac{10 - 4}{2} = 3 , \text{inches} \]
Now, we form a right triangle with the height (4 inches) and half the difference between the bases (3 inches). The length of the non-parallel side \( l \) can be calculated using the Pythagorean theorem:
\[ l = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 , \text{inches} \]
Each pair of lateral faces corresponds to the lengths of the two non-parallel sides (5 inches each), plus the two parallel sides of the trapezoid (10 inches and 4 inches). Therefore, the total lateral area is calculated as follows:
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Area of the two lateral rectangles corresponding to the non-parallel sides: \[ 2 \cdot l \cdot h = 2 \cdot 5 \cdot 10 = 100 , \text{in}^2 \]
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Area of the two lateral rectangles corresponding to the bases: \[ \text{Area for 10 in base} = 10 \cdot 10 = 100 , \text{in}^2 \] \[ \text{Area for 4 in base} = 4 \cdot 10 = 40 , \text{in}^2 \]
Adding the above:
\[ \text{Total surface area} = 56 , \text{in}^2 + 100 , \text{in}^2 + 100 , \text{in}^2 + 40 , \text{in}^2 = 296 , \text{in}^2 \]
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Therefore, the correct answer is 296 in².
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