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.

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

2)

Step-by-step explanation:

1) Here we have;

First tank A

Volume of tank = x³

The  volume of the sphere =

However, the diameter of the sphere = x therefore;

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

volume of the sphere = =

For tank B

Volume of tank = x³

The  volume of the spheres =

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 =

For tank C

Volume of tank = x³

The  volume of the spheres =

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 =

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 =

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 =

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 =

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