Question
Two pyramids like the one below sit on top of the gates at the entrance to the bridge.
A square pyramid. The base is 10 inches by 10 inches. The triangular sides have a base of 10 inches and height of 13 inches. The pyramid has a height of 12 inches.
Which expression shows how to determine the minimum amount of cement used to make each of the pyramids?
One-third (10) (10) (12)
One-half (10) (10) (12)
4 (one-half (10) (13)) + 10 (10)
4 (one-third (10) (13)) + 10 (10)
A square pyramid. The base is 10 inches by 10 inches. The triangular sides have a base of 10 inches and height of 13 inches. The pyramid has a height of 12 inches.
Which expression shows how to determine the minimum amount of cement used to make each of the pyramids?
One-third (10) (10) (12)
One-half (10) (10) (12)
4 (one-half (10) (13)) + 10 (10)
4 (one-third (10) (13)) + 10 (10)
Answers
GPT-4o mini
To determine the minimum amount of cement used to make each pyramid, we need to calculate both the volume of the pyramid and the area of the triangular sides.
1. **Volume of the pyramid**: The volume \( V \) of a pyramid can be calculated using the formula:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
For a square pyramid:
- Base Area = \( 10 \times 10 = 100 \, \text{square inches} \)
- Height = 12 inches
Therefore, the volume of one pyramid is:
\[
V = \frac{1}{3} \times 100 \times 12 = \frac{1200}{3} = 400 \, \text{cubic inches}
\]
2. **Surface area of the triangular sides**: The surface area of the triangular sides consists of 4 triangular faces. The area \( A \) of one triangular face is given by:
\[
A = \frac{1}{2} \times \text{Base} \times \text{Height}
\]
Here, the base of each triangle is 10 inches and the height is 13 inches:
\[
A = \frac{1}{2} \times 10 \times 13 = 65 \, \text{square inches}
\]
Since there are 4 triangular faces, the total area for the triangular sides is:
\[
\text{Total Area} = 4 \times 65 = 260 \, \text{square inches}
\]
Given these calculations, the expression to find the minimum amount of cement would primarily focus on the volume when considering how much material is needed to create the structural body of the pyramid, while also acknowledging the surface area of the sides.
Among the provided options, none directly specifies a formula for combining volume and surface area in one expression; however, the relevant terms appear in:
- **One-third (10)(10)(12)** for the volume calculation of the pyramid.
- **One-half (10)(13)** times 4 for the sides.
The closest interpretation to the details provided, particularly considering the importance of both the triangular face calculations and the base, could be expressed as either:
- \( 4 \left( \frac{1}{2} (10)(13) \right) + 10(10) \) but this seems misplaced since the base is not being treated correctly.
Thus, the clearest expression that indicates the way to calculate the volume would be:
- **One-third (10)(10)(12)** illustrates how to determine the cement for just the volume of the pyramids, which seems like a more straightforward answer for an expression that represents a concrete minimum amount of material for the structure itself.
Hence, the best choice that represents the calculation for cement used for each pyramid is:
**One-third (10)(10)(12)**.
1. **Volume of the pyramid**: The volume \( V \) of a pyramid can be calculated using the formula:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
For a square pyramid:
- Base Area = \( 10 \times 10 = 100 \, \text{square inches} \)
- Height = 12 inches
Therefore, the volume of one pyramid is:
\[
V = \frac{1}{3} \times 100 \times 12 = \frac{1200}{3} = 400 \, \text{cubic inches}
\]
2. **Surface area of the triangular sides**: The surface area of the triangular sides consists of 4 triangular faces. The area \( A \) of one triangular face is given by:
\[
A = \frac{1}{2} \times \text{Base} \times \text{Height}
\]
Here, the base of each triangle is 10 inches and the height is 13 inches:
\[
A = \frac{1}{2} \times 10 \times 13 = 65 \, \text{square inches}
\]
Since there are 4 triangular faces, the total area for the triangular sides is:
\[
\text{Total Area} = 4 \times 65 = 260 \, \text{square inches}
\]
Given these calculations, the expression to find the minimum amount of cement would primarily focus on the volume when considering how much material is needed to create the structural body of the pyramid, while also acknowledging the surface area of the sides.
Among the provided options, none directly specifies a formula for combining volume and surface area in one expression; however, the relevant terms appear in:
- **One-third (10)(10)(12)** for the volume calculation of the pyramid.
- **One-half (10)(13)** times 4 for the sides.
The closest interpretation to the details provided, particularly considering the importance of both the triangular face calculations and the base, could be expressed as either:
- \( 4 \left( \frac{1}{2} (10)(13) \right) + 10(10) \) but this seems misplaced since the base is not being treated correctly.
Thus, the clearest expression that indicates the way to calculate the volume would be:
- **One-third (10)(10)(12)** illustrates how to determine the cement for just the volume of the pyramids, which seems like a more straightforward answer for an expression that represents a concrete minimum amount of material for the structure itself.
Hence, the best choice that represents the calculation for cement used for each pyramid is:
**One-third (10)(10)(12)**.