Question
Latoria has a cube and a square-based pyramid. The two figures have the same height and their bases have the same area
Answers
Answered by
GPT-4o mini
To analyze the relationship between the cube and the square-based pyramid that Latoria has, let’s denote the height of both the cube and the pyramid as \( h \).
### Cube:
1. **Volume of the Cube**:
- If the side length of the cube is \( s \), then the volume \( V \) of the cube is given by:
\[
V_{\text{cube}} = s^3
\]
2. **Base Area of the Cube**:
- The base area \( A \) of the cube (which is a square) is:
\[
A_{\text{cube}} = s^2
\]
### Square-Based Pyramid:
1. **Volume of the Pyramid**:
- If the base side length of the pyramid is \( b \), then the volume \( V \) of the pyramid is given by:
\[
V_{\text{pyramid}} = \frac{1}{3} b^2 h
\]
2. **Base Area of the Pyramid**:
- The base area \( A \) of the pyramid is:
\[
A_{\text{pyramid}} = b^2
\]
### Given Conditions:
- Both figures have the same height: \( h \).
- The bases have the same area:
\[
A_{\text{cube}} = A_{\text{pyramid}} \implies s^2 = b^2
\]
From the equality of base areas, we can deduce that:
\[
s = b
\]
since both \( s \) and \( b \) are lengths and must be non-negative.
### Volumes:
By substituting \( b = s \) into the volume formula for the pyramid, we get:
\[
V_{\text{pyramid}} = \frac{1}{3} s^2 h
\]
Now we can summarize the volumes:
- Volume of the Cube:
\[
V_{\text{cube}} = s^3
\]
- Volume of the Pyramid:
\[
V_{\text{pyramid}} = \frac{1}{3} s^2 h
\]
### Relations Between Their Volumes:
To compare the volumes of the two shapes, we can write:
\[
V_{\text{pyramid}} = \frac{1}{3} A_{\text{cube}} h
\]
Since \( A_{\text{cube}} = s^2 \), we can see:
\[
V_{\text{pyramid}} = \frac{1}{3} s^2 h
\]
Therefore, comparing both:
- Volume of the Cube \( V_{\text{cube}} \) is three times the volume of the pyramid when you solve for height \( h \):
\[
V_{\text{cube}} = 3 V_{\text{pyramid}}
\]
In conclusion, Latoria's cube has a volume that is three times greater than the volume of her square-based pyramid, provided they have the same height and their bases share the same area.
### Cube:
1. **Volume of the Cube**:
- If the side length of the cube is \( s \), then the volume \( V \) of the cube is given by:
\[
V_{\text{cube}} = s^3
\]
2. **Base Area of the Cube**:
- The base area \( A \) of the cube (which is a square) is:
\[
A_{\text{cube}} = s^2
\]
### Square-Based Pyramid:
1. **Volume of the Pyramid**:
- If the base side length of the pyramid is \( b \), then the volume \( V \) of the pyramid is given by:
\[
V_{\text{pyramid}} = \frac{1}{3} b^2 h
\]
2. **Base Area of the Pyramid**:
- The base area \( A \) of the pyramid is:
\[
A_{\text{pyramid}} = b^2
\]
### Given Conditions:
- Both figures have the same height: \( h \).
- The bases have the same area:
\[
A_{\text{cube}} = A_{\text{pyramid}} \implies s^2 = b^2
\]
From the equality of base areas, we can deduce that:
\[
s = b
\]
since both \( s \) and \( b \) are lengths and must be non-negative.
### Volumes:
By substituting \( b = s \) into the volume formula for the pyramid, we get:
\[
V_{\text{pyramid}} = \frac{1}{3} s^2 h
\]
Now we can summarize the volumes:
- Volume of the Cube:
\[
V_{\text{cube}} = s^3
\]
- Volume of the Pyramid:
\[
V_{\text{pyramid}} = \frac{1}{3} s^2 h
\]
### Relations Between Their Volumes:
To compare the volumes of the two shapes, we can write:
\[
V_{\text{pyramid}} = \frac{1}{3} A_{\text{cube}} h
\]
Since \( A_{\text{cube}} = s^2 \), we can see:
\[
V_{\text{pyramid}} = \frac{1}{3} s^2 h
\]
Therefore, comparing both:
- Volume of the Cube \( V_{\text{cube}} \) is three times the volume of the pyramid when you solve for height \( h \):
\[
V_{\text{cube}} = 3 V_{\text{pyramid}}
\]
In conclusion, Latoria's cube has a volume that is three times greater than the volume of her square-based pyramid, provided they have the same height and their bases share the same area.