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 24.1 mm and a thickness of 1.78 mm. and it is completely covered with a layer of pure gold 0.180 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 $25.0 per gram, (a) Find the cost of the gold added to the coin, (b) Does the cost of the gold significantly enhance the value of the coin? Explain your answer.
(a)
The cost of gold added to the coin.
Answer to Problem 1.54AP
The cost of gold added to the coin is
Explanation of Solution
Given info: The thickness of the coin is
Write the expression for total surface area of the coin,
Here,
Formula to calculate the area of curved surface of the coin is,
Here,
Formula to calculate the area of faces of the coin is,
Substitute
Substitute
Thus, the total surface area is
Write the expression for volume of the gold plating,
Here,
Substitute
Thus, the volume of gold plating is
Formula to calculate the mass of the gold used for plating is,
The density of the gold is
Substitute
Thus, the mass of gold plating on the coin is
Write the expression to calculate the cost of gold added to the coin,
Substitute
Conclusion:
Therefore, the cost of gold added to the coin is
(b)
The percentage the cost of the gold significantly enhance the value of the coin.
Answer to Problem 1.54AP
The cost of gold significantly enhances the value of the cost of coin by
Explanation of Solution
Given info: The thickness of the coin is
From the part (a), the cost of gold added to the coin is
The value of the coin is quarter dollar
Write the expression for enhance value,
Substitute
Conclusion:
Therefore, the cost of gold significantly enhances the value of the cost of coin by
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