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 areas of the rectangular lateral faces.
-
Calculate the area of the trapezoidal base: The formula for the area \(A\) of a trapezoid is:
\[ A = \frac{1}{2} \times (b_1 + b_2) \times h \]
where \(b_1\) and \(b_2\) are the lengths of the two bases, and \(h\) is the height of the trapezoid.
Here, \(b_1 = 4\) in, \(b_2 = 10\) in, and \(h = 4\) in.
\[ A = \frac{1}{2} \times (4 + 10) \times 4 = \frac{1}{2} \times 14 \times 4 = 28 \text{ in}^2 \]
-
Calculate the area of the two trapezoidal bases: The prism has two bases, so we need to multiply the area of one trapezoid by 2:
\[ \text{Total area of trapezoidal bases} = 2 \times 28 = 56 \text{ in}^2 \]
-
Calculate the area of the lateral faces: The lateral faces of the prism are rectangles. There are 4 lateral faces for a prism, corresponding to each side of the trapezoid.
- The lengths of the two non-parallel sides can be determined using the height of the trapezoid, which is the vertical height between the two bases, and the difference in lengths of the bases.
- Since the trapezoid is isosceles, we can drop perpendiculars from the endpoints of the shorter base to form two right triangles. The length of the segment connecting the perpendicular foot to the midpoint of the shorter base is:
\[ \frac{10 - 4}{2} = 3 \text{ in} \]
Thus, the length of each non-parallel side can be found using the Pythagorean theorem:
\[ \text{Side length} = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ in} \]
The prism has the following lateral faces:
- Two rectangles corresponding to the lengths of the bases, each of dimensions \(10 \times 10\) and \(4 \times 10\).
- Two rectangles corresponding to the non-parallel sides (which are both 5 in), each of dimension \(5 \times 10\).
Now, we calculate the areas:
- Area of the rectangle with base 10 in: \( 10 \times 10 = 100 \text{ in}^2 \)
- Area of the rectangle with base 4 in: \( 4 \times 10 = 40 \text{ in}^2 \)
- Area of the two rectangles from non-parallel sides: \( 2 \times (5 \times 10) = 2 \times 50 = 100 \text{ in}^2 \)
Total area of all lateral faces:
\[ \text{Total area of lateral faces} = 100 + 40 + 100 = 240 \text{ in}^2 \]
-
Add the areas of the bases and the lateral faces:
\[ \text{Total surface area} = \text{Area of bases} + \text{Area of lateral faces} = 56 + 240 = 296 \text{ in}^2 \]
Thus, the correct answer is 296 in\(^2\).