Question

You are relaxing at home, working on your coin collection. You have learned that collectible coins are sometimes plated with gold to enhance their beauty and value. On an online site that sells collectible coins, you see a commemorative quarter-dollar advertised for sale at $4.98. The information on the site says that the coin has a diameter of 24.6 mm and a thickness of 2.08 mm, and it is completely covered with a layer of pure gold 0.165 µm thick. You know that gold is a valuable asset and you come up with a scheme by which you would separate the gold from the base coin and sell the gold at the current market price. You begin your scheme by buying the advertised coin at $4.98. (Assume the density of gold is 19.3 g/cm3.)

What is the value of the gold for a single such coin (in cents)? Assume that the price of gold is $33.00 per gram.?

What If? What would be the cost (in dollars) of the gold to produce the coin if the coin were made entirely of gold ?

Answer 1

R = radius of the coin = diameter/2 = 24.6 mm/2 = 12.3 mm = 0.0123 m

H = thickness of the coin = 2.08 mm = 0.00208 m

V_c = Volume of the coin

Volume of the coin is given as

V_c = $\pi$R² H

t = thickness of gold = 0.165 x 10^-6 m

r = radius of coin without gold = R - t = 0.0123 - 0.165 x 10^-6

h = thickness of the coin without gold = H - 2t = 0.00208 - 2 ( 0.165 x 10^-6 )

V' = volume of the coin without gold

volume of the coin without gold is given as

V' = $\pi$r² h

v = volume of gold

Volume of gold is given as

v = V_c - V'

v = $\pi$R² H - $\pi$r² h

v = $\pi$(0.0123)² (0.00208) - $\pi$(0.0123 - 0.165 x 10^-6)² (0.00208 - 2 ( 0.165 x 10^-6 ))

v = 1.83365 x 10^-10

$\rho$ = density of gold = 19.3 g/cm³ = 19.3 x 10³ kg/m³

m = mass of gold

mass of gold is given as

m = $\rho$ v

m = (19.3 x 10³) (1.83365 x 10^-10)

m = 3.54 x 10^-6 kg = 3.54 x 10^-3 g

c = cost per gram of gold = $33

C = total cost of gold on coin

C = m c

C = 3.54 x 10^-3 ($33 )

C = $ 0.117

C = 11.7 cents

v'' = volume of gold required for the coin to be made of gold

volume of gold required is given as

v'' = $\pi$R² H

v'' = $\pi$(0.0123)² (0.00208)

v'' = 9.89 x 10^-7 m³

m'' = mass of gold

m'' = $\rho$ v''

m'' = (19.3 x 10³) (9.89 x 10^-7)

m'' = 0.01909 kg = 19.09 g

Cost is given as

C = m'' c

C = 19.09 x $33

C = $630