Question
Latoria has a cube and a square-based pyramid. The two figures have the same height and their bases have the same area. Latoria knows that the volume of the cube is 330 cubic inches. What is the volume of the pyramid?(1 point)The volume of the pyramid is in.3.
Answers
Answered by
GPT-4o mini
To find the volume of the pyramid, we need to remember the formula for the volume of a square-based pyramid and the relationship between the cube and the pyramid based on their height and base area.
1. **Volume of the Cube**: The volume of the cube (V_cube) is given as 330 cubic inches. The volume of a cube is calculated using the formula:
\[
V_{cube} = s^3
\]
where \( s \) is the length of one side of the cube.
To find \( s \):
\[
s^3 = 330 \implies s = \sqrt[3]{330}
\]
2. **Volume of the Pyramid**: The volume of a square-based pyramid (V_pyramid) is given by:
\[
V_{pyramid} = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
Since the cube and the pyramid have the same height and their bases have the same area, let’s denote:
- Base Area = A
- Height = h
Given that the pyramid's base area is equal to the cube's base area, we can express the base area in terms of the cube. The area of the base of the cube (which is a square) is:
\[
A = s^2 = (\sqrt[3]{330})^2 = 330^{2/3}
\]
The height of the cube (which is also the height of the pyramid) is \( s = \sqrt[3]{330} \).
3. **Using the Formula for Volume of the Pyramid**:
Plugging in the values we have:
\[
V_{pyramid} = \frac{1}{3} \times A \times h = \frac{1}{3} \times 330^{2/3} \times \sqrt[3]{330}
\]
The height \( h \) of the pyramid is \( \sqrt[3]{330} \), so:
\[
V_{pyramid} = \frac{1}{3} \times 330^{2/3} \times 330^{1/3} = \frac{1}{3} \times 330^{(2/3 + 1/3)} = \frac{1}{3} \times 330^{1} = \frac{330}{3} = 110
\]
Thus, the volume of the pyramid is \( \boxed{110} \) cubic inches.
1. **Volume of the Cube**: The volume of the cube (V_cube) is given as 330 cubic inches. The volume of a cube is calculated using the formula:
\[
V_{cube} = s^3
\]
where \( s \) is the length of one side of the cube.
To find \( s \):
\[
s^3 = 330 \implies s = \sqrt[3]{330}
\]
2. **Volume of the Pyramid**: The volume of a square-based pyramid (V_pyramid) is given by:
\[
V_{pyramid} = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
Since the cube and the pyramid have the same height and their bases have the same area, let’s denote:
- Base Area = A
- Height = h
Given that the pyramid's base area is equal to the cube's base area, we can express the base area in terms of the cube. The area of the base of the cube (which is a square) is:
\[
A = s^2 = (\sqrt[3]{330})^2 = 330^{2/3}
\]
The height of the cube (which is also the height of the pyramid) is \( s = \sqrt[3]{330} \).
3. **Using the Formula for Volume of the Pyramid**:
Plugging in the values we have:
\[
V_{pyramid} = \frac{1}{3} \times A \times h = \frac{1}{3} \times 330^{2/3} \times \sqrt[3]{330}
\]
The height \( h \) of the pyramid is \( \sqrt[3]{330} \), so:
\[
V_{pyramid} = \frac{1}{3} \times 330^{2/3} \times 330^{1/3} = \frac{1}{3} \times 330^{(2/3 + 1/3)} = \frac{1}{3} \times 330^{1} = \frac{330}{3} = 110
\]
Thus, the volume of the pyramid is \( \boxed{110} \) cubic inches.