A rectangular prism has a volume of 27 in cubic if a rectangular pyramid has a base and height congruent to the prism what is the volume of the pyramid
Since the pyramid has a base and height congruent to the prism, it means that the base of the pyramid is also a rectangle with the same dimensions as the base of the prism.
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 prism is 27 cubic inches, it means that:
27 = lwh.
Since the base of the pyramid has the same dimensions as the base of the prism, the length and width of the base are the same. Therefore, let's say the length and width of the base of the pyramid are x inches each. This means that:
27 = x * x * h
27 = x^2 * h
h = 27 / x^2.
Since the height of the pyramid is also congruent to the height of the prism, the height of the pyramid is the same as the height of the prism. Therefore, the volume of the pyramid can be calculated using the formula for the volume of a pyramid, which is V = (1/3) * B * h, where B is the base area of the pyramid and h is the height of the pyramid.
Since the base of the pyramid is a rectangle with sides x and x, the base area is x^2. Therefore, the volume of the pyramid is:
V = (1/3) * x^2 * (27 / x^2)
V = (1/3) * 27
V = 9 cubic inches.
Therefore, the volume of the pyramid is 9 cubic inches.