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.

\( radius = \frac{41 \text{ mm}}{2} = 20.5 \text{ mm} \) #tag_title#Step 2: Calculate the volume of the coin#tag_content#Using the radius and thickness of the coin, we can find the volume using the formula for the volume of a cylinder: \( V = \pi r^2h \) where V is the volume, r is the radius, and h is the thickness. The height in this case is the thickness of the coin, which is 2.5 mm. \( V = \pi (20.5 \text{ mm})^2 (2.5 \text{ mm}) = 3330.15 \text{ mm}^3 \) Now convert the volume to cm³: \( 1 \text{ cm}^3 = 1000 \text{ mm}^3 \) \( V = 3330.15 \text{ mm}^3 \times \frac{1 \text{ cm}^3}{1000 \text{ mm}^3} = 3.33015 \text{ cm}^3 \) #tag_title#Step 3: Calculate the mass of the coin#tag_content#Using the density and the volume of the coin, we can calculate its mass: \( mass = density \times volume \) The density of silver is given as 10.5 g/cm³: \( mass = (10.5 \, \text{g}/\text{cm}^3) \times 3.33015 \, \text{cm}^3 = 34.966575 \, \text{g} \) #tag_title#Step 4: Calculate the value of the silver in the coin#tag_content#Finally, we multiply the mass of the coin by the market price of silver per gram to find the value of the silver in the coin: \( Value = mass \times price \) The market price of silver is given as $0.51 per gram: \( Value = 34.966575 \, \text{g} \times \$0.51/\text{g} = \$17.83317375 \)