 Construct a cuboid with 24 small cubes, as shown opposite.

The surface area of the cuboid is

2 x (2 x 3 + 4 x 2 + 4 x 3) = 52 square units where the colours are top and sides

Does the surface area change if the arrangement (configuration) of cubes changes? Problem 1   How many cubes are needed for the cuboid opposite?

Problem 2   Find the surface area of this cuboid.

Problem 3   Using 24 cubes to make a cuboid, what is the configuration that gives

a) minimum surface area

b) maximum surface area?

The minimum surface area configuration corresponds to the shape that requires the minimum amount of wrapping paper for a given volume (i.e. 24 cubic units).  Of course, overlaps would be needed in practice, but as they would be similar for all shapes, we can disregard them hee. Problem 4   Find the minimum wrapping for

a) 48 cubes                 b) 64 cubes.

A related problem is to find the total amount of string required to go round the cuboid in each direction (as shown opposite), not including the extra needed for the knot.

Problem 5   For the cuboids you made earlier from 24 cubes, find the total length of string needed for each configuration.

Problem 6   What configuration minimises the amount of string needed to tie up a parcel of volume 24 cubic units? EXTENSION

1.   Find the cuboid shape that minimises the surface area when it encloses a fixed volume, V.

2.   Does this also minimse the string needed?