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 $$4\mathrm{.}98.$ The information on the site says that the coin has a diameter of $24\mathrm{.}6$ mm and a thickness of $2\mathrm{.}07$ mm$,$ and it is completely covered with a layer of pure gold $0\mathrm{.}160$m thick. You know that gold is a valuable asset and you come up with a scheme by which you would separate the gold from the base coin and sell the gold at the current market price. You begin your scheme by buying the advertised coin at $$4\mathrm{.}98.($Assume the density of gold is $19\mathrm{.}3$ g$/$cm$3.)$

$($a$)$

What is the value of the gold for a single such coin $($in cents$)?$ Assume that the price of gold is $$33\mathrm{.}00$ per gram.

$($b$)$

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

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