The U.S. penny was originally minted entirely of copper....

The U.S. penny was originally minted entirely of copper. However, after the year 1857 the U.S. penny was made of a variety of metals because of the high cost of copper. Since 1982 pennies have been made of zinc plated with a thin layer of copper. The modern penny weighs $2.500$ grams, has a diameter of $19.05$ millimeters, and an average thickness of $1.224$ millimeters. Given that the density of copper and zinc are $8.96 \mathrm{~g} \cdot \mathrm{cm}^{-3}$ and $7.13 \mathrm{~g} \cdot \mathrm{cm}^{-3}$, respectively, determine the percentage by mass of copper in a modern penny.

Key Concepts

Density

Density is a fundamental physical property defined as mass per unit volume. It allows us to relate the mass of a substance to the volume it occupies and is expressed in units such as grams per cubic centimeter. In problems involving materials, knowing the density helps determine how much mass is contributed by a given volume of the material.

Volume Calculation

Volume calculation involves determining the space occupied by an object based on its geometric dimensions. For regular shapes like cylinders, the volume can be computed with formulas such as V = ?r²h. This concept is critical when calculating the total volume of a coin or any object, particularly when converting between different units of measurement.

Composite Materials

Composite materials are those made from two or more constituent materials with different physical or chemical properties. When combined, these materials create a new material with characteristics different from the individual components. In problems involving composites, it is important to determine the contribution of each component to the overall mass and volume based on their individual properties such as density.

Percentage by Mass

Percentage by mass is a measure that expresses the mass of a component as a fraction of the total mass of a mixture or composite, multiplied by 100. This concept helps in understanding the composition of an alloy or composite material by quantifying the proportion of each constituent present within the overall mass.

Transcript

00:01 We have to find the percent by mass of copper in a penny given the penny's mass, diameter, radius, thickness, and the density of the two most prominent metals copper and zinc.

00:14 So our first step is to find the volume of the penny.

00:18 So we're going to use the volume of a cylinder formula because a penny is just a very very thin cylinder.

00:25 So we're going to take pi times the radius squared times its thickness or its height and we're going to get 348 .868 cubic millimeters.

00:38 Next we have to convert this to cubic centimeters because that is what our density is given in.

00:45 So we're going to take 3 .48 .868 and we're going to divide it by 0 .1000.

00:52 This is just 10 times 10 times 10.

00:56 So we're just converting from millimeters to centimeters three different times because we have three different dimensions.

01:02 So this is going to end up being 0 .3488 and then we'll just say 6 .8 and then that is cubic centimeters.

01:15 So now let's convert our penny's mass to a density.

01:21 So from here, we're going to say the density, the total density rather of our penny is 2 .500 grams over 0 .348868 cubic centimeters.

01:40 So we'll take 2 .500 and divide it by our previous answer and we're going to get 7 .166 grams per cubic centimeter.

01:54 So as we can see that it's really really close to the density of zinc.

02:00 So the majority of our penny is going to be zinc...

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