+1833 A spherical holiday ornament with a radius r is to be packacked in a cubical box so that all sides touch the sphere. What is the ratio of the volume of the sphere to the volume of the cube? If the same ornament in the problem above is to be packed in a cylinder so that all surfaces touch the sphere, what is the ratio of the volume of the sphere to the volume of the cylinder?
+128408 Good to see you, Mellie....!!!
V cube = s^3 where s is the side length of the cube (1)
V sphere in a cube = (4/3)pi (1/2s)^3 (2)
Ratio of (2) to (1) =
(4/3)pi (1/2s)^3 / s^3 =
(4/3)pi *(1/8)s^3 /s^3
(4/3)pi (1/8) / 1 =
pi / 6
V cylinder = pi * r^2 * h → pi*r^2 *(2 r ) → 2* pi*r^3 (1)
V sphere = (4/3) * pi * r^3 (2)
Ratio of (2) to (1) =
[(4/3) pi * r^3 ] / [ 2 *pi * r^3 ] =
(4/3) / 2 =
4/6 =
2/3
