To find the maximum volume of a square pyramid that can fit inside the cube, we need to determine the dimensions of the square base and the height of the pyramid.
Since the cube has a side length of 15cm, the diagonal of the base of the square pyramid cannot exceed 15cm. This means that the side length of the base of the square pyramid must be less than or equal to 15cm. To maximize the volume of the pyramid, we should have the maximum possible side length for the base.
Let's assume that the side length of the base of the square pyramid is equal to 15cm. In this case, the diagonal of the base would be:
diagonal = side length * sqrt(2)
diagonal = 15 * sqrt(2)
diagonal ≈ 21.21 cm
This means that the height of the pyramid would be equal to:
height = sqrt(diagonal^2 - ((side length / 2)^2))
height = sqrt(21.21^2 - (7.5^2))
height = sqrt(451 - 56.25)
height = sqrt(394.75)
height ≈ 19.87 cm
Therefore, the maximum volume of the square pyramid that can fit inside the cube is given by:
V = (1/3) * (side length)^2 * height
V = (1/3) * 15^2 * 19.87
V = (1/3) * 225 * 19.87
V = 1/3 * 4478.75
V ≈ 1492.92 cm^3
Therefore, the maximum volume of the square pyramid that can fit inside the cube is approximately 1492.92 cm^3.
There is a cube with a side length of 15cm. What is the maximum volume of a square pyramid that can fit inside the cube?
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1 answer