A cube with volume 8 cubic cm is inscribed in a sphere so that each vertex of the cube touches the sphere. What is the length of the diameter, in centimeters, of the sphere?
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The diameter of the sphere in this case will be equal to the long diagonal of the cube, i.e., the diagonal between the vertices of the cube that are furthest apart. To determine the length of the diagonal we'll need to find the length of the hypotenuse of a right triangle with one short side being an edge of the cube and the other short side being the diagonal across one of the cubes faces. As the volume of the sphere is 8 cm^3, each edge will be 8^(1/3) = 2 cm long, and so the face diagonal will be
sqrt(2^2 + 2^2) = 2sqrt(2). Thus the long diagonal will be
(long diagonal) = sqrt((edge length)^2 + (face diagonal)^2) =
sqrt(2^2 + (2sqrt(2))^2) = sqrt(4 + 8) = sqrt(12) = 2sqrt(3) cm = 3.46 cm, which will also be the diameter of the sphere.
sqrt(2^2 + 2^2) = 2sqrt(2). Thus the long diagonal will be
(long diagonal) = sqrt((edge length)^2 + (face diagonal)^2) =
sqrt(2^2 + (2sqrt(2))^2) = sqrt(4 + 8) = sqrt(12) = 2sqrt(3) cm = 3.46 cm, which will also be the diameter of the sphere.
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8 = s^3
2 = s
2^2 = 3d^2
2^2 / 3 = d^2
2 * sqrt(3) / 3 = d
2 = s
2^2 = 3d^2
2^2 / 3 = d^2
2 * sqrt(3) / 3 = d