Question
Which prism has a greater surface area?
2 prisms. A rectangular prism has a length of 12 inches, height of 8 inches, and width of 6 inches. A triangular prism has a rectangular base with a length of 6 inches and height of 12 inches. 2 rectangular sides are 12 inches by 10 inches. The triangular sides have a base of 6 inches an height of 8 inches.
The rectangular prism has a greater surface area by 72 square inches.
The rectangular prism has a greater surface area by 88 square inches.
The triangular prism has a greater surface area by 72 square inches.
The triangular prism has a greater surface area by 88 square inches.
2 prisms. A rectangular prism has a length of 12 inches, height of 8 inches, and width of 6 inches. A triangular prism has a rectangular base with a length of 6 inches and height of 12 inches. 2 rectangular sides are 12 inches by 10 inches. The triangular sides have a base of 6 inches an height of 8 inches.
The rectangular prism has a greater surface area by 72 square inches.
The rectangular prism has a greater surface area by 88 square inches.
The triangular prism has a greater surface area by 72 square inches.
The triangular prism has a greater surface area by 88 square inches.
Answers
GPT-4o mini
To determine which prism has a greater surface area, we'll calculate the surface area of both the rectangular prism and the triangular prism.
### Rectangular Prism
The formula for the surface area \(SA\) of a rectangular prism is:
\[
SA = 2lw + 2lh + 2wh
\]
Where \(l\) is the length, \(w\) is the width, and \(h\) is the height.
Substituting the values:
- Length \(l = 12\) inches
- Width \(w = 6\) inches
- Height \(h = 8\) inches
\[
SA = 2(12 \times 6) + 2(12 \times 8) + 2(6 \times 8)
\]
Calculating each term:
- \(2(12 \times 6) = 2(72) = 144\)
- \(2(12 \times 8) = 2(96) = 192\)
- \(2(6 \times 8) = 2(48) = 96\)
Adding them together:
\[
SA = 144 + 192 + 96 = 432 \text{ square inches}
\]
### Triangular Prism
The formula for the surface area of a triangular prism is:
\[
SA = (base_{area} \times height) + (perimeter_{triangular base} \times height)
\]
Where the base area can be found using:
\[
base_{area} = \frac{1}{2} \times base \times height_{triangular}
\]
For our triangular prism:
- Base of triangle = 6 inches
- Height of triangle = 8 inches
- Height of prism = 12 inches
Calculating the area of the triangular base:
\[
base_{area} = \frac{1}{2} \times 6 \times 8 = 24 \text{ square inches}
\]
Calculating the perimeter of the triangular base:
The triangles have a base of 6 inches and the two sides are both 10 inches tall. So the perimeter is:
\[
perimeter = 6 + 10 + 10 = 26 \text{ inches}
\]
Now substituting into the surface area formula:
\[
SA_{triangular} = (24 \times 12) + (26 \times 12)
\]
Calculating each term:
- \(24 \times 12 = 288\)
- \(26 \times 12 = 312\)
Adding them together:
\[
SA_{triangular} = 288 + 312 = 600 \text{ square inches}
\]
### Comparison
Now, we compare the surface areas:
- Rectangular Prism Surface Area: 432 square inches
- Triangular Prism Surface Area: 600 square inches
The triangular prism has a greater surface area.
To find the difference in surface area:
\[
600 - 432 = 168 \text{ square inches}
\]
### Conclusion
The triangular prism has a greater surface area by 168 square inches. However, this specific option does not appear in your list of choices. If the options don't include this answer, please check the dimensions or consider if there may be a calculation or data error.
### Rectangular Prism
The formula for the surface area \(SA\) of a rectangular prism is:
\[
SA = 2lw + 2lh + 2wh
\]
Where \(l\) is the length, \(w\) is the width, and \(h\) is the height.
Substituting the values:
- Length \(l = 12\) inches
- Width \(w = 6\) inches
- Height \(h = 8\) inches
\[
SA = 2(12 \times 6) + 2(12 \times 8) + 2(6 \times 8)
\]
Calculating each term:
- \(2(12 \times 6) = 2(72) = 144\)
- \(2(12 \times 8) = 2(96) = 192\)
- \(2(6 \times 8) = 2(48) = 96\)
Adding them together:
\[
SA = 144 + 192 + 96 = 432 \text{ square inches}
\]
### Triangular Prism
The formula for the surface area of a triangular prism is:
\[
SA = (base_{area} \times height) + (perimeter_{triangular base} \times height)
\]
Where the base area can be found using:
\[
base_{area} = \frac{1}{2} \times base \times height_{triangular}
\]
For our triangular prism:
- Base of triangle = 6 inches
- Height of triangle = 8 inches
- Height of prism = 12 inches
Calculating the area of the triangular base:
\[
base_{area} = \frac{1}{2} \times 6 \times 8 = 24 \text{ square inches}
\]
Calculating the perimeter of the triangular base:
The triangles have a base of 6 inches and the two sides are both 10 inches tall. So the perimeter is:
\[
perimeter = 6 + 10 + 10 = 26 \text{ inches}
\]
Now substituting into the surface area formula:
\[
SA_{triangular} = (24 \times 12) + (26 \times 12)
\]
Calculating each term:
- \(24 \times 12 = 288\)
- \(26 \times 12 = 312\)
Adding them together:
\[
SA_{triangular} = 288 + 312 = 600 \text{ square inches}
\]
### Comparison
Now, we compare the surface areas:
- Rectangular Prism Surface Area: 432 square inches
- Triangular Prism Surface Area: 600 square inches
The triangular prism has a greater surface area.
To find the difference in surface area:
\[
600 - 432 = 168 \text{ square inches}
\]
### Conclusion
The triangular prism has a greater surface area by 168 square inches. However, this specific option does not appear in your list of choices. If the options don't include this answer, please check the dimensions or consider if there may be a calculation or data error.