Chapter 1, Problem 1.54AP

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.

To determine

The cost of gold added to the coin.

Answer to Problem 1.54AP

The cost of gold added to the coin is $\$0.091$.

Explanation of Solution

Given info: The thickness of the coin is $1.78\text{mm}$, the diameter of the coin is $24.1\text{mm}$, the thickness of pure gold coating on the coin is $0.180\text{μm}$ and the price of gold is $\$25.0$ per gram.

Write the expression for total surface area of the coin,

$A=A_{\text{C}}+A_{\text{F}}$ (1)

Here,

$A_{\text{C}}$ is the curved surface area of the coin.

$A_{\text{F}}$ is the area of the face of coin.

Formula to calculate the area of curved surface of the coin is,

$A_{\text{C}}=\pi dh$

Here,

$h$ is the thickness of the coin.

$d$ is the diameter of the coin.

Formula to calculate the area of faces of the coin is,

$A_{\text{F}}=2\left(\frac{\pi d^2}{4}\right)$

Substitute $2\left(\frac{\pi d^2}{4}\right)$ for $A_{\text{F}}$ and $\pi dh$ for $A_{\text{C}}$ in the equation (1).

$A=\left[\pi dh+2\left(\frac{\pi d^2}{4}\right)\right]$

Substitute $24.1\text{mm}$ for $d$ and $1.78\text{mm}$ for $h$ in the above equation.

$\begin{array}{c}A=\left[\begin{array}{l}\pi \left(24.1\text{mm}\right)\left(\frac{1\text{m}}{1000\text{mm}}\right)\left(1.78\text{mm}\right)\left(\frac{1\text{m}}{1000\text{mm}}\right)+\\ \frac{2\pi }{4}{\left(\left(24.1\text{mm}\right)\left(\frac{1\text{m}}{1000\text{mm}}\right)\right)}^2\end{array}\right]\\ =1.35\times {10}^{-4}{\text{m}}^2+9.12\times {10}^{-4}{\text{m}}^2\\ =10.47\times {10}^{-4}{\text{m}}^2\end{array}$

Thus, the total surface area is $10.47\times {10}^{-4}{\text{m}}^2$.

Write the expression for volume of the gold plating,

$V_{\text{gold}}=t_{\text{gold}}A$ (2)

Here,

$t_{\text{gold}}$ is the thickness of gold plating layer on the coin.

Substitute $0.180\text{μm}$ for $t_{\text{gold}}$ and $10.47\times {10}^{-4}{\text{m}}^2$ for $A$ in the equation (2).

$\begin{array}{c}{V}_{\text{gold}}=\left(0.180\text{μm}\right)\left(10.47\times {10}^{-4}{\text{m}}^2\right)\\ =\left(0.180\times {10}^{-6}\text{m}\right)\left(10.47\times {10}^{-4}{\text{m}}^2\right)\\ =1.8846\times {10}^{-10}{\text{m}}^3\end{array}$

Thus, the volume of gold plating is $1.8846\times {10}^{-10}{\text{m}}^3$.

Formula to calculate the mass of the gold used for plating is,

$m_{\text{gold}}={\rho }_{\text{gold}}{V}_{\text{gold}}$ (3)

The density of the gold is $19,300\text{kg}/{\text{m}}^{\text{3}}$.

Substitute $19,300\text{kg}/{\text{m}}^{\text{3}}$ for ${\rho }_{\text{gold}}$ and $1.8846\times {10}^{-10}{\text{m}}^3$ for $V_{\text{gold}}$ in the equation (3).

$\begin{array}{c}{m}_{\text{gold}}=\left(19,300\text{kg}/{\text{m}}^{\text{3}}\right)\left(1.8846\times {10}^{-10}{\text{m}}^3\right)\\ =3.637\times {10}^{-6}\text{kg}\left(\frac{{10}^3\text{g}}{1\text{kg}}\right)\\ =0.003637\text{g}\end{array}$

Thus, the mass of gold plating on the coin is $0.003637\text{g}$.

Write the expression to calculate the cost of gold added to the coin,

$\text{cost}\text{of}\text{gold}=\left(m_{\text{gold}}\right)\left(\text{price}\text{of}\text{gold}\right)$

Substitute $0.003637\text{g}$ for $\left(m_{\text{gold}}\right)$ and $\$25.0$ for price of gold in the above equation.

$\begin{array}{c}\text{cost}\text{of}\text{gold}=\left(0.003637\text{g}\right)\left(\frac{\$25.0}{\text{g}}\right)\\ =\$0.091\end{array}$

Conclusion:

Therefore, the cost of gold added to the coin is $\$0.091$.

To determine

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 $36.4\%$ , when gold plating is added to the coin.

Explanation of Solution

Given info: The thickness of the coin is $1.78\text{mm}$, the diameter of the coin is $24.1\text{mm}$, the thickness of pure gold coating on the coin is $0.180\text{μm}$ and the price of gold is $\$25.0$ per gram.

From the part (a), the cost of gold added to the coin is $\$0.091$.

The value of the coin is quarter dollar $\left(\$0.25\right)$.

Write the expression for enhance value,

$\text{Enhance}\text{value}\left(\%\right)=\frac{\text{value}\text{added}}{\text{real}\text{value}}\times 100$ (5)

Substitute $\$0.091$ for value added and $\$0.25$ for real value in the equation (5).

$\begin{array}{c}\text{Enhance}\text{value}\left(\%\right)=\frac{\$0.091}{\$0.25}\times 100\%\\ =36.4\%\end{array}$

Conclusion:

Therefore, the cost of gold significantly enhances the value of the cost of coin by $36.4\%$ , when gold plating is added to the coin.