Question

A prisoner of a pyramid both have a triangular base if both shapes have the same height in the volume of the prism is 600 ft.³ what is the volume of the pyramid?

Answer 1

To find the volume of the pyramid, we first need to find the area of the triangular base of the prism. Since the prism and pyramid both have the same height, the base of the pyramid is also a triangle with the same dimensions as the base of the prism.

Let's assume the base of the triangle has a length of x ft. The area of the base of the prism is given by:

Area of triangular base = 1/2 * base * height = 1/2 * x * height = 1/2 * x * x = x^2/2

Since the volume of the prism is 600 ft^3, we can set up the following equation:

Volume of prism = Area of base * height = 600
x^2/2 * height = 600

We know that the height is the same for both the prism and the pyramid, so we can use the same height in the pyramid.

The volume of the pyramid is given by:

Volume of pyramid = 1/3 * base * height = 1/3 * x * x * height = 1/3 * x^2 * height

From the equation of the prism, we can solve for x using: x^2/2 * height = 600
x^2/2 * height = 600
x^2 = 2 * 600
x = sqrt(2 * 600) = sqrt(1200) = 20 * sqrt(3)

Therefore, the volume of the pyramid is:
Volume of pyramid = 1/3 * (20 * sqrt(3))^2 * height
Volume of pyramid = 1/3 * 400 * 3 * height
Volume of pyramid = 400 * height
Volume of pyramid = 400 * height

Since the height is the same for both the prism and the pyramid, the volume of the pyramid is 400 ft^3.