To find the volume of a rectangular prism that has a congruent base and height twice that of a rectangular pyramid, we first need to understand how volume is calculated for both shapes.
-
Volume of a Pyramid: The formula for the volume \( V \) of a pyramid is given as: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Given that the volume of the pyramid is \( 10 , \text{cm}^3 \): \[ 10 = \frac{1}{3} \times \text{Base Area} \times h \]
-
Volume of a Prism: The formula for the volume \( V \) of a rectangular prism is: \[ V = \text{Base Area} \times \text{Height} \]
-
Congruent Base and Height: Since the rectangular prism has the same base area as the pyramid and a height that is twice that of the pyramid's height (\( 2h \)): \[ V_{\text{prism}} = \text{Base Area} \times (2h) \]
-
Relating the Volumes: We can express the volume of the prism in terms of the pyramid volume: \[ V_{\text{prism}} = \text{Base Area} \times (2h) = 2 \times \left( \frac{1}{3} \times \text{Base Area} \times h \right) = 2 \times 10 = 20 , \text{cm}^3 \]
Therefore, the volume of the rectangular prism is \( \boxed{20 , \text{cm}^3} \).