The U.S. Mint produces a dollar coin called the American Silver Eagle that is made of nearly pure silver. This coin has a diameter of \(41 \mathrm{~mm}\) and a thickness of \(2.5 \mathrm{~mm}\). The density and approximate market price of silver are \(10.5 \mathrm{~g} / \mathrm{cm}^{3}\) and \(\$ 0.51\) per gram, respectively. Calculate the value of the silver in the coin, assuming its thickness is uniform.

The American Silver Eagle coin is cylindrical in shape. The formula to calculate the volume of a cylinder is: \[V = \pi r^2 h\] where: - \(V\) is the volume of the cylinder, - \(r\) is the radius (which is half of the diameter), - \(h\) is the height (or thickness in this case), - and \(\pi\) (pi) is a mathematical constant approximately equal to \(3.14159\). Given the diameter of the coin is \(41 \mathrm{~mm}\), the radius can be determined as: \[r = \frac{41 \mathrm{~mm}}{2} = 20.5 \mathrm{~mm}\] Moreover, the thickness of the coin is \(2.5 \mathrm{~mm}\). On substituting the values in the formula, we will get the volume of the coin in cubic millimeters: \[V = \pi (20.5 \mathrm{~mm})^2 (2.5 \mathrm{~mm})\]

Since the given density and price are in terms of cubic centimeters and grams, respectively, we should convert the volume from cubic millimeters to cubic centimeters. To do this conversion, we use the following formula: \[1 \mathrm{~cm}^3 = 1000 \mathrm{~mm}^3\] So, to convert the volume from cubic millimeters to cubic centimeters: \[V_\mathrm{cm^3} = V_\mathrm{mm^3} \times \frac{1 \mathrm{~cm}^3}{1000 \mathrm{~mm}^3}\]

Now that we have the volume in cubic centimeters, we can use the density of silver \(10.5 \mathrm{~g/cm}^3\) to obtain the mass of the coin. The following formula can be used: \[M = V_\mathrm{cm^3} \times D\] where: - \(M\) is the mass of the coin, - \(V_\mathrm{cm^3}\) is the volume of the coin in cubic centimeters, - and \(D\) is the density of silver (\(10.5 \mathrm{~g/cm}^3\)).

Finally, we can calculate the value of the silver in the coin using the mass obtained in step 3 and the given price of silver (\(\$ 0.51\) per gram). Using the following formula, we can find the value of the silver: \[Value = M \times P\] where: - \(Value\) is the value of the silver in the coin, - \(M\) is the mass of the coin, - and \(P\) is the price per gram of silver (\(\$ 0.51\)).