Note: it is not 100% clear whether the US Mint charges a premium on the gold content (0.9167 ounces) as the analysis above assumes or on the total weight of the coin (1.000 ounces). If the later is indeed the case it would add $4 to the US Mint cost (roughly 3% on 8%).
The 'Coin Dealer' cost data is from a popular internet dealer. Dealer commissions vary and are known to change over time.
From the US Mint:
American Eagle Gold Bullion Coins
Public Law 99-185, enacted December 17, 1985, directs the United States Mint to mint and issue legal tender gold bullion coins. The coins are .9167 fine gold (22 karat), with the following weights and face values: Weight – troy oz. Face value of fine gold (denomination) 1 oz. $50 1/2 oz. $25 1/4 oz. $10 1/10 oz. $5 (In addition to the .9167 gold, the coins are composed of .0300 silver and .0533 copper.)
For the Gold Eagles, we charge 3%, 5%, 7% and 9% premiums for the one, one-half, one-quarter and one-tenth ounce coins respectively. Minimum ordering requirements are 1000 ounces.
The pricing of the precious metal content is established at the time of the sale: The London P.M. (second) Gold Fix on the date following the day of the order, excluding federal government holidays. The fixed premiums are a percentage of the gold price: 3 percent, 5 percent, 7 percent and 9 percent for the 1, 1/2, 1/4 and 1/10 ounce coins, respectively.
Purchased coins are authorized for release after receipt of payment is confirmed. All United States Mint gold bullion coins must be picked up freight-on-board (F.O.B.) at the United States Mint at West Point (West Point, New York).