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Chapter 2: Problem 103

A U.S. 1 -cent coin (a penny) has a diameter of 19 \(\mathrm{mm}\) and a thickness of 1.5 \(\mathrm{mm}\) . Assume the coin is made of pure copper, whose density and approximate market price are 8.9 \(\mathrm{g} / \mathrm{cm}^{3}\) and \(\$ 2.40\) per pound, respectively. Calculate the value of the copper in the coin, assuming its thickness is uniform.

Short Answer

Expert verified
The value of the copper in a U.S. 1-cent coin (a penny) is approximately $0.008.

Step by step solution

01

Calculate the volume of the coin

Since the penny is a cylinder with uniform thickness, we can use the formula for the volume of a cylinder: Volume = π * (radius)^2 * height Here, the diameter is 19 mm, so the radius is 9.5 mm, and the thickness (height) is 1.5 mm. Converting to centimeters, we get radius = 0.95 cm and height = 0.15 cm. Now, we can calculate the volume: Volume = π * (0.95)^2 * 0.15 cm^3
02

Calculate the mass of the coin

We are given the density of copper as 8.9 g/cm^3. To find the mass of the copper in the penny, we need to multiply the volume by the density: Mass = Density * Volume = (8.9 g/cm^3) * (π * (0.95)^2 * 0.15 cm^3)
03

Calculate the value of the copper

The value of the copper in the penny can be determined using the given price of copper: $2.40 per pound. First, we need to convert the mass from grams to pounds: Mass (pounds) = Mass (grams) * (1 pound / 453.592 g) Now, we can calculate the value of the copper in the penny: Value = Mass (pounds) * Price per pound Value = (Mass (grams) * (1 pound / 453.592 g)) * $2.40 per pound Combining all the steps and substituting the given variables: Value = ((8.9 g/cm^3) * (π * (0.95)^2 * 0.15 cm^3) * (1 pound / 453.592 g)) * $2.40 per pound Calculating the value: Value ≈ $0.008

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