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

**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),

**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?

**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?