Question
Surface Area of a Model Practice Complete this assessment to review what you’ve learned. It will not count toward your grade. 1 of 51 of 5 Items Question Determine the surface area of a prism with a triangular base. The base is an isosceles triangle with a base length of 12 in. and a height of 8 in. The height of the prism is 14 in.(1 point) in.2 Remaining Attempts : 3 Skip to navigation
Answers
GPT-4o mini
To determine the surface area of a prism with a triangular base, first we need to calculate the surface area of the triangular base and then include the lateral surface area of the prism.
1. **Calculate the area of the base triangle:**
- The area \( A \) of a triangle is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
- For the isosceles triangle, the base length is 12 in. and the height is 8 in.:
\[
A = \frac{1}{2} \times 12 \, \text{in} \times 8 \, \text{in} = \frac{1}{2} \times 96 \, \text{in}^2 = 48 \, \text{in}^2
\]
2. **Calculate the height of the triangular sides (lateral faces) of the prism:**
- The perimeter of the triangular base is needed to find the lateral surface area.
- To find the lengths of the two equal sides of the isosceles triangle, we can use the Pythagorean theorem. The half-base is \( \frac{12}{2} = 6 \) in., and the height is 8 in. Thus, the length of each of the equal sides \( s \) is calculated as follows:
\[
s = \sqrt{(6 \, \text{in})^2 + (8 \, \text{in})^2} = \sqrt{36 \, \text{in}^2 + 64 \, \text{in}^2} = \sqrt{100 \, \text{in}^2} = 10 \, \text{in}
\]
- The perimeter \( P \) of the triangular base is:
\[
P = 12 \, \text{in} + 10 \, \text{in} + 10 \, \text{in} = 32 \, \text{in}
\]
3. **Calculate the lateral surface area:**
- The lateral surface area \( A_{lateral} \) of the prism is given by the formula:
\[
A_{lateral} = \text{Perimeter} \times \text{Height of Prism}
\]
- Given the height of the prism is 14 in.:
\[
A_{lateral} = 32 \, \text{in} \times 14 \, \text{in} = 448 \, \text{in}^2
\]
4. **Calculate the total surface area:**
- The total surface area (SA) of the prism consists of the areas of the two triangular bases and the lateral surface area:
\[
SA = 2 \times A + A_{lateral}
\]
- Substitute the values we found:
\[
SA = 2 \times 48 \, \text{in}^2 + 448 \, \text{in}^2 = 96 \, \text{in}^2 + 448 \, \text{in}^2 = 544 \, \text{in}^2
\]
Thus, the total surface area of the prism is **544 in²**.
1. **Calculate the area of the base triangle:**
- The area \( A \) of a triangle is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
- For the isosceles triangle, the base length is 12 in. and the height is 8 in.:
\[
A = \frac{1}{2} \times 12 \, \text{in} \times 8 \, \text{in} = \frac{1}{2} \times 96 \, \text{in}^2 = 48 \, \text{in}^2
\]
2. **Calculate the height of the triangular sides (lateral faces) of the prism:**
- The perimeter of the triangular base is needed to find the lateral surface area.
- To find the lengths of the two equal sides of the isosceles triangle, we can use the Pythagorean theorem. The half-base is \( \frac{12}{2} = 6 \) in., and the height is 8 in. Thus, the length of each of the equal sides \( s \) is calculated as follows:
\[
s = \sqrt{(6 \, \text{in})^2 + (8 \, \text{in})^2} = \sqrt{36 \, \text{in}^2 + 64 \, \text{in}^2} = \sqrt{100 \, \text{in}^2} = 10 \, \text{in}
\]
- The perimeter \( P \) of the triangular base is:
\[
P = 12 \, \text{in} + 10 \, \text{in} + 10 \, \text{in} = 32 \, \text{in}
\]
3. **Calculate the lateral surface area:**
- The lateral surface area \( A_{lateral} \) of the prism is given by the formula:
\[
A_{lateral} = \text{Perimeter} \times \text{Height of Prism}
\]
- Given the height of the prism is 14 in.:
\[
A_{lateral} = 32 \, \text{in} \times 14 \, \text{in} = 448 \, \text{in}^2
\]
4. **Calculate the total surface area:**
- The total surface area (SA) of the prism consists of the areas of the two triangular bases and the lateral surface area:
\[
SA = 2 \times A + A_{lateral}
\]
- Substitute the values we found:
\[
SA = 2 \times 48 \, \text{in}^2 + 448 \, \text{in}^2 = 96 \, \text{in}^2 + 448 \, \text{in}^2 = 544 \, \text{in}^2
\]
Thus, the total surface area of the prism is **544 in²**.