Collectible coins are sometimes plated with gold to enhance...

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 $\mathrm{mm}$ and a thickness of $1.78 \mathrm{mm},$ and it is completely covered with a layer of pure gold 0.180$\mu \mathrm{m}$ thick. The volume of the plating is equal to the thickness of the layer times 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 $\$ 10.0$ per gram. Find the cost of the gold added to the coin. Does the cost of the gold significantly enhance the value of the coin? Explain your answer.

Key Concepts

Economic Value Evaluation

Economic value evaluation involves comparing the calculated cost of material inputs to the overall market price of an item. This helps determine if the material cost is a major contributor to the item's value or if other factors, like rarity or design, play a more significant role in its market premium.

Material Cost Analysis

This concept ties the physical characteristics of a material, such as its volume (and, if needed, density), to its monetary value by using a price per unit mass or volume. It is essential for evaluating the cost-effectiveness of using specific materials in manufacturing or art, by linking quantitative calculations to economic decisions.

Geometry and Surface Area Calculation

This concept involves determining the total surface area of an object—in this case, a coin—which can include the areas of its flat faces as well as its edge. Accurate surface area calculation is essential when dealing with coatings or layers applied uniformly over an object.

Volume Calculation for Thin Layers

When a thin material layer, such as a plating, is applied to a surface, its volume can be approximated by multiplying the surface area by the thickness of the layer. This approach is useful in various applications, from engineering to materials science, where knowing the volume of material used is necessary for further analysis.

Unit Conversion

Unit conversion is the process of converting measurements from one unit to another to maintain consistency across calculations. For example, converting micrometers to millimeters or centimeters ensures that the dimensions used in calculations are coherent and accurate, which is critical in precise scientific and engineering applications.

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