To determine whether Shannon is correct, we first need to understand the volumes of the rectangular pyramid and the rectangular prism.
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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): \[ 60 = \frac{5}{3} \times \text{BA} \ 60 \times 3 = 5 \times \text{BA} \ 180 = 5 \times \text{BA} \ \text{BA} = \frac{180}{5} = 36 \text{ units}^2 \]
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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: \[ V = 36 \times 5 = 180 \text{ units}^3 \]
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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."
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