A man makes complete use of 405 cc of iron, 783 cc of aluminium, and 351 cc of copper to make a number of solid right circular cylinders of each type of metal. These cylinders have the same volume and each of these have radius 3 cm. If the total number of cylinders is to be kept at a minimum, then the total surface area of all these cylinders, in sq cm, is

To minimize the number of cylinders, the volume of each cylinder must be the greatest common divisor (GCD) of the metal volumes: 405, 783, and 351, which is 27 cc. With a radius of 3 cm, the height of each cylinder is h = 3/pi cm. The total number of cylinders is 57, and multiplying this by the surface area of a single cylinder, 18(1 + pi), gives a total surface area of 1026(1 + pi) sq cm.