# Problem of the Week ## Rectangular Light

This light fitting is made from a series of 4 open rectangular metal prisms.

The smallest rectangular prism has an open base that is 20 cm long x 10 cm deep.

The next largest prism has an open base that is always 2 cm longer on each side and 2 cm deeper on each side than the previous prism in the series.

If each rectangular prism has a height of 5 cm, what area of metal sheeting is needed to make this light fitting?

The metal required for the 4 rectangular prisms could be calculated separately, but because they all have the same height, it is easier to calculate the perimeters of each base, add these lengths together and multiply the total length by the height of 5 cm.

The perimeter of the smallest prism = 20 cm + 10 cm + 20 cm + 10 cm

= 2 x (20 + 10) cm

= 2 x 30 cm

= 60 cm

The perimeter of the next smallest prism = 2 x (24 + 14) cm

= 2 x 38 cm

= 76 cm

The perimeter of the next smallest prism = 2 x (28 + 18) cm

= 2 x 46 cm

= 92 cm

The perimeter of the largest prism = 2 x (32 + 22) cm

= 2 x 54 cm

= 108 cm

(Note that the perimeter increases by 16 each time, because 2 cm x 2 x 2 x 2 = 16 cm)

Total of the 4 perimeters = 60 cm + 76 cm + 92 cm + 108 cm = 336 cm

So, the area of metal sheet needed = 5 cm (height) x 336 cm (total length) = 1680 cm2