Question

In the illustration below, the three cube-shaped tanks are identical. The spheres in any given tank

are the same size and packed wall-to-wall. If each of the tanks are filled to the top with water, which
tank would contain the most water. Prove your answer algebraically using x to represent the edge
length of the tanks.

Write an expression to represent the amount of water remaining in a 4th tank which is the same size
as the others and which contains 512 spheres. Leave your expression in terms of π.

Answer 1

1) Volume occupied by the spheres are equal therefore the three tanks contains the same volume of water

2)
`Amount \ of \, water \ remaining \ in \, the \ tank \ is \ (x^3(6-\pi) )/(6)`

Explanation:

1) Here we have;

First tank A

Volume of tank = x³

The volume of the sphere =
`(4)/(3) \pi r^3`

However, the diameter of the sphere = x therefore;

r = x/2 and the volume of the sphere is thus;

volume of the sphere =
`(4)/(3) \pi (x^3)/(8)`=
`(1)/(6) \pi x^3`

For tank B

Volume of tank = x³

The volume of the spheres =
`8 * (4)/(3) \pi r^3`

However, the diameter of the spheres 2·D = x therefore;

r = x/4 and the volume of the sphere is thus;

volume of the spheres =
`8 * (4)/(3) \pi ((x)/(4))^3= (x^3 * \pi )/(6)`

For tank C

Volume of tank = x³

The volume of the spheres =
`64 * (4)/(3) \pi r^3`

However, the diameter of the spheres 4·D = x therefore;

r = x/8 and the volume of the sphere is thus;

volume of the spheres =
`64 * (4)/(3) \pi ((x)/(8))^3= (x^3 * \pi )/(6)`

Volume occupied by the spheres are equal therefore the three tanks contains the same volume of water

2) For the 4th tank, we have;

number of spheres on side of the tank, n is given thus;

n³ = 512

∴ n = ∛512 = 8

Hence we have;

Volume of tank = x³

The volume of the spheres =
`512 * (4)/(3) \pi r^3`

However, the diameter of the spheres 8·D = x therefore;

r = x/16 and the volume of the sphere is thus;

volume of the spheres =
`512* (4)/(3) \pi ((x)/(16))^3= (x^3 * \pi )/(6)`

Amount of water remaining in the tank is given by the following expression;

Amount of water remaining in the tank = Volume of tank - volume of spheres

Amount of water remaining in the tank =
`x^3 - (x^3 * \pi )/(6) = (x^3(6-\pi) )/(6)`

`Amount \ of \ water \, remaining \, in \, the \ tank = (x^3(6-\pi) )/(6)`.