Question
A rectangular pyramid has a height of 5 units and a volume of 60 units3. Shannon states that a rectangular prism with the same base area and height has a volume that is three times the size of the given rectangular pyramid. Which statement explains whether Shannon is correct?
A rectangular prism in which BA = 36 and h = 5 has a volume of 180 units3; therefore, Shannon is correct
A rectangular prism in which BA = 12 and h = 5 has a volume of 180 units3; therefore, Shannon is correct
A rectangular prism in which BA = 36 and h = 5 has a volume of 60 units3; therefore, Shannon is incorrect
A rectangular prism in which BA = 12 and h = 5 has a volume of 60 units3; therefore, Shannon is incorrect
A rectangular prism in which BA = 36 and h = 5 has a volume of 180 units3; therefore, Shannon is correct
A rectangular prism in which BA = 12 and h = 5 has a volume of 180 units3; therefore, Shannon is correct
A rectangular prism in which BA = 36 and h = 5 has a volume of 60 units3; therefore, Shannon is incorrect
A rectangular prism in which BA = 12 and h = 5 has a volume of 60 units3; therefore, Shannon is incorrect
Answers
GPT-4o mini
To determine whether Shannon is correct, we first need to understand the volumes of the rectangular pyramid and the rectangular prism.
1. **Volume of a Rectangular Pyramid:**
The formula for the volume \( V \) of a rectangular pyramid is given by:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
We know the volume of the pyramid is 60 units³ and the height is 5 units. Thus, we can set up the equation:
\[
60 = \frac{1}{3} \times \text{Base Area} \times 5
\]
Solving for the Base Area (BA):
\[
60 = \frac{5}{3} \times \text{BA} \\
60 \times 3 = 5 \times \text{BA} \\
180 = 5 \times \text{BA} \\
\text{BA} = \frac{180}{5} = 36 \text{ units}^2
\]
2. **Volume of a Rectangular Prism:**
The formula for the volume \( V \) of a rectangular prism is given by:
\[
V = \text{Base Area} \times \text{Height}
\]
If the rectangular prism has the same Base Area = 36 units² and height = 5 units, then the volume of the prism is:
\[
V = 36 \times 5 = 180 \text{ units}^3
\]
3. **Comparison with Shannon's Statement:**
Shannon states that a rectangular prism with the same base area and height has a volume that is three times the size of the pyramid. Since the volume of the pyramid is 60 units³:
\[
3 \times \text{Volume of Pyramid} = 3 \times 60 = 180 \text{ units}^3
\]
Therefore, Shannon is correct because the volume of the rectangular prism, given it has the same base area of 36 units² and height of 5 units, is indeed 180 units³, which is three times the volume of the pyramid.
**Correct Statement:**
- "A rectangular prism in which BA = 36 and h = 5 has a volume of 180 units³; therefore, Shannon is correct."
1. **Volume of a Rectangular Pyramid:**
The formula for the volume \( V \) of a rectangular pyramid is given by:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
We know the volume of the pyramid is 60 units³ and the height is 5 units. Thus, we can set up the equation:
\[
60 = \frac{1}{3} \times \text{Base Area} \times 5
\]
Solving for the Base Area (BA):
\[
60 = \frac{5}{3} \times \text{BA} \\
60 \times 3 = 5 \times \text{BA} \\
180 = 5 \times \text{BA} \\
\text{BA} = \frac{180}{5} = 36 \text{ units}^2
\]
2. **Volume of a Rectangular Prism:**
The formula for the volume \( V \) of a rectangular prism is given by:
\[
V = \text{Base Area} \times \text{Height}
\]
If the rectangular prism has the same Base Area = 36 units² and height = 5 units, then the volume of the prism is:
\[
V = 36 \times 5 = 180 \text{ units}^3
\]
3. **Comparison with Shannon's Statement:**
Shannon states that a rectangular prism with the same base area and height has a volume that is three times the size of the pyramid. Since the volume of the pyramid is 60 units³:
\[
3 \times \text{Volume of Pyramid} = 3 \times 60 = 180 \text{ units}^3
\]
Therefore, Shannon is correct because the volume of the rectangular prism, given it has the same base area of 36 units² and height of 5 units, is indeed 180 units³, which is three times the volume of the pyramid.
**Correct Statement:**
- "A rectangular prism in which BA = 36 and h = 5 has a volume of 180 units³; therefore, Shannon is correct."