What is the maximum volume of a square pyramid that can fit inside a cube with a side length of 18 cm?
(1 point)
Responses
5,832 cm3
5,832 cm 3
2,916 cm3
2,916 cm 3
1,944 cm3
1,944 cm 3
972 cm3
The maximum volume of a square pyramid that can fit inside a cube is when the base of the pyramid is the same as the base of the cube. Since the cube has a side length of 18 cm, the base of the pyramid would also have a side length of 18 cm.
To find the maximum volume, we can use the formula for the volume of a pyramid: V = (1/3) * base area * height.
The base area of the pyramid is 18 cm * 18 cm = 324 cm^2.
To find the height of the pyramid, we can use the Pythagorean theorem. The height, h, is the slant height of the pyramid, and it can be found using the formula h = sqrt(l^2 - b^2), where l is the length of one side of the pyramid's base and b is the length of the square that connects the midpoints of the base's sides. In this case, l = 18 cm and b = (18/2) = 9 cm. Plugging these values into the formula, we get h = sqrt(18^2 - 9^2) = sqrt(324 - 81) = sqrt(243) = 15.59 cm (rounded to two decimal places).
Now we can find the volume of the pyramid: V = (1/3) * 324 cm^2 * 15.59 cm ≈ 1624.85 cm^3 (rounded to two decimal places).
Therefore, the maximum volume of a square pyramid that can fit inside a cube with a side length of 18 cm is approximately 1624.85 cm^3.