To find the maximum volume of a square pyramid that can fit inside a cube, we first need to determine the dimensions of the square pyramid.
Since the base of the pyramid is a square, its side length will be equal to that of the cube, which is 18 inches. The height of the pyramid is not given, but we know that the height of the pyramid will be less than the height of the cube (as the pyramid needs to fit inside the cube).
Let's denote the height of the pyramid as h inches.
The formula for the volume of a square pyramid is given by V = (1/3) * base area * height
The base area of the square pyramid is the area of the square base, which is equal to the side length squared, so it is 18^2 = 324 square inches.
Therefore, the volume of the square pyramid can be expressed as V = (1/3) * 324 * h = 108h cubic inches.
Since the pyramid must fit within the cube, the height of the pyramid must be less than 18 inches. Let's denote the maximum height of the pyramid that would allow it to fit inside the cube as x.
Therefore, the maximum volume of the square pyramid that can fit inside the cube would be 108x cubic inches.
To find the maximum volume of the pyramid, we set x = 18 inches:
V = 108 * 18 = 1944 cubic inches
Therefore, the maximum volume of a square pyramid that can fit inside a cube with a side length of 18 inches is 1944 cubic inches.
What is the maximum volume of a square pyramid that can fit inside a cube with a side length of 18 inches.
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