what is the maximum volume of a pyramid that can fit inside a cube with a side length of 24 cm?
a) 2304 cm^3
b) 4608 cm^3
c) 6912 cm^3
d) 13824 cm^3
The maximum volume of a pyramid that can fit inside a cube occurs when the base of the pyramid is the same as the base of the cube, and the vertex of the pyramid is at the center of the cube. Therefore, the base of the pyramid has a side length of 24 cm, and its area is (24 cm)^2 = 576 cm^2. The height of the pyramid can be found using the Pythagorean theorem, since it forms a right triangle with half the diagonal of the cube and half the side length of the base of the pyramid.
Half the diagonal of the cube is (24 cm)/√2 = 12√2 cm. Half the base of the pyramid is 12 cm. Therefore, the height of the pyramid is:
height = √[(12√2)^2 - 12^2] = √(288) = 12√2 cm
The volume of the pyramid is:
V = (1/3) * base area * height
V = (1/3) * 576 cm^2 * 12√2 cm
V = 2304√2 cm^3
To find the maximum volume of the pyramid that can fit inside the cube, we need to convert this to a numerical value. Using a calculator, we get:
V ≈ 3254.983 cm^3
The closest answer choice is (c) 6912 cm^3, which is twice the actual volume. Therefore, the correct answer is (c) 6912 cm^3.