Question

Extra Credit:

8 congruent spheres are packed into a cube with
edge length x so that each sphere is tangent to 3
faces of the cube and 3 other spheres as shown.
what is the ratio of the total volume of the
spheres of the volume of the cube?

Answer 1

Ratio of the volumes of the spheres and cube = 0.523

Explanation:

From the figure attached,

Diameter of one sphere = Half of the measure of one side of the cube

=
`(x)/(2)`

Radius of the sphere =
`((x)/(2))/(2)`

=
`(x)/(4)`

Volume of a cube is given by the formula,

V =
`(4)/(3)\pi r^(3)`

Therefore, volume of one sphere =
`(4)/(3)\pi ((x)/(4))^(3)`

=
`(x^3\pi)/(48)`

Volume of 8 spheres =
`8* (x^3\pi)/(48)`

=
`(x^3\pi)/(6)`

Volume of a cube = (side)³

=
`x^3`

Ratio of the volumes of the sphere and cube =
`((\pi x^3)/(6) )/(x^3)`

=
`(\pi)/(6)`

≈ 0.523