Question
If X = 7 units, Y = 12 units, Z = 16 units, and h = 4 units, what is the surface area of the triangular prism
A.
384 square units
B.
512 square units
C.
464 square units
D.
363 square units
Answers
GPT-4o mini
To find the surface area of a triangular prism, we need to calculate the areas of the two triangular bases and the three rectangular sides.
1. **Area of the triangular base**:
For a triangle, the area can be calculated using the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, we'll take \(X\) as the base, which is 7 units, and \(h\) as the height of the triangle, which is 4 units.
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times 7 \times 4 = \frac{28}{2} = 14 \text{ square units}
\]
Since there are two triangular bases:
\[
\text{Total area of triangular bases} = 2 \times 14 = 28 \text{ square units}
\]
2. **Area of the rectangular sides**:
The three rectangular sides will have dimensions based on the lengths of the sides of the triangular base and the height (h) of the prism.
- Side 1 (base X = 7 units):
\[
\text{Area} = \text{base} \times h = 7 \times 4 = 28 \text{ square units}
\]
- Side 2 (base Y = 12 units):
\[
\text{Area} = 12 \times 4 = 48 \text{ square units}
\]
- Side 3 (base Z = 16 units):
\[
\text{Area} = 16 \times 4 = 64 \text{ square units}
\]
3. **Total area of rectangular sides**:
\[
\text{Total area} = 28 + 48 + 64 = 140 \text{ square units}
\]
4. **Total surface area of the prism**:
\[
\text{Surface Area} = \text{Total area of triangular bases} + \text{Total area of rectangular sides}
\]
\[
\text{Surface Area} = 28 + 140 = 168 \text{ square units}
\]
However, the calculated surface area of the triangular prism (168 square units) is not matching with any of the provided multiple-choice options. Please verify the values or formulas used, as there might be some additional dimensions or alternate data affecting the calculations.
1. **Area of the triangular base**:
For a triangle, the area can be calculated using the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, we'll take \(X\) as the base, which is 7 units, and \(h\) as the height of the triangle, which is 4 units.
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times 7 \times 4 = \frac{28}{2} = 14 \text{ square units}
\]
Since there are two triangular bases:
\[
\text{Total area of triangular bases} = 2 \times 14 = 28 \text{ square units}
\]
2. **Area of the rectangular sides**:
The three rectangular sides will have dimensions based on the lengths of the sides of the triangular base and the height (h) of the prism.
- Side 1 (base X = 7 units):
\[
\text{Area} = \text{base} \times h = 7 \times 4 = 28 \text{ square units}
\]
- Side 2 (base Y = 12 units):
\[
\text{Area} = 12 \times 4 = 48 \text{ square units}
\]
- Side 3 (base Z = 16 units):
\[
\text{Area} = 16 \times 4 = 64 \text{ square units}
\]
3. **Total area of rectangular sides**:
\[
\text{Total area} = 28 + 48 + 64 = 140 \text{ square units}
\]
4. **Total surface area of the prism**:
\[
\text{Surface Area} = \text{Total area of triangular bases} + \text{Total area of rectangular sides}
\]
\[
\text{Surface Area} = 28 + 140 = 168 \text{ square units}
\]
However, the calculated surface area of the triangular prism (168 square units) is not matching with any of the provided multiple-choice options. Please verify the values or formulas used, as there might be some additional dimensions or alternate data affecting the calculations.