Collectible coins are sometimes plated with gold to enhance their beauty and val
ID: 1950245 • Letter: C
Question
Collectible coins are sometimes plated with gold to enhance their beauty and value. Consider a commemorative quarter-dollar advertised for sale at $4.98. It has a diameter of 24.3 mm, a thickness of 1.78 mm, and is completely covered with a layer of pure gold 0.139 µm thick. The volume of the plating is equal to the thickness of the layer multiplied by the area to which it is applied. The patterns on the faces of the coin and the grooves on its edge have a negligible effect on its area. Assume the price of gold is $24.0 per gram.Find the cost of the gold added to the coin.
Explanation / Answer
.139e-6 layer of gold 24.3e-3 diameter 1.78e-3 thickness Surface area of coin is the sum of the areas of the top, the bottom, and the side. The top and bottom combined have an area of 2*pi*r^2 = (24.3e-3/2)^2*pi*2 = 7.63e-2. The area of the side is the circumference times the thickness of the coin (think of it as a rectangle) and that is pi*diameter*thickness = pi*(2.43e-3) * 1.78e-3 = 8.65e5. We add the two areas for a total surface area of 7.64e-2. The volume of plating is then 7.64e-2 * .139e-6 = 1.06e-8. The density of gold is 19320 kg/m^3, so we have 19320 * 1.06e-8 = 2.05e-4 kg = 0.205 g. The cost per gram is given, so total cost is .205*24.00 = $4.93.