Question

A base of a prism is an isosceles trapezoid that has one base length of 4 in., another base length of 10 in., and a height of 4 in. The height of the prism is 10 in. Which of the following correctly calculates the surface area of the prism with an isosceles trapezoid base?(1 point)
Responses

220 in2
220 in. squared

296 in2
296 in. squared

432 in2
432 in. squared

300 in2

Answer 1

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.

1. **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:

\[
A = \frac{1}{2} \times (4 + 10) \times 4 = \frac{1}{2} \times 14 \times 4 = 28 \, \text{in}^2
\]

2. **Area of the two trapezoidal bases:**
Since the prism has two bases, we multiply the area of one base by 2:

\[
\text{Total area of the bases} = 28 \, \text{in}^2 \times 2 = 56 \, \text{in}^2
\]

3. **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:

- Area of the two lateral rectangles corresponding to the non-parallel sides:
\[
2 \cdot l \cdot h = 2 \cdot 5 \cdot 10 = 100 \, \text{in}^2
\]

- 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:

\[
\text{Total surface area} = 56 \, \text{in}^2 + 100 \, \text{in}^2 + 100 \, \text{in}^2 + 40 \, \text{in}^2 = 296 \, \text{in}^2
\]

Therefore, the correct answer is **296 in²**.