A rectangular prism has a volume of 252 cubed inches. If a rectangular pyramid has a base and height congruent to the prism, what is the volume of the pyramid?
The volume of a rectangular prism is given by the formula: V = lwh, where l is the length, w is the width, and h is the height.
Given that the volume of the rectangular prism is 252 cubic inches, we can calculate the dimensions of the prism using the formula:
252 = lwh
Since the base and height of the rectangular pyramid are congruent to the prism, we can assume that the base of the pyramid is a square with sides of length l and the height of the pyramid is also equal to h.
Therefore, the volume of the pyramid can be calculated using the formula for the volume of a pyramid: V = (1/3) Bh, where B is the area of the base and h is the height.
Since the base is a square with sides of length l, the area of the base B is given by: B = l^2
Therefore, the volume of the pyramid is: V = (1/3)(l^2)(h)
Now we can substitute the values of l and h from the prism into the formula for the volume of the pyramid:
V = (1/3)(l^2)(h) = (1/3)(l^2)(l) = (1/3)(l^3)
Since the volume of the rectangular prism is 252 cubic inches, we know that l^3 = 252
Taking the cube root of both sides gives: l = 6
Therefore, the volume of the pyramid is: V = (1/3)(6^3) = (1/3)(216) = 72 cubic inches
So, the volume of the rectangular pyramid is 72 cubic inches.