Problem 103 - Chemistry The Central Science

Question

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.

Step-by-Step Solution

Answer

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

Step 1: 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

Step 2: 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)

Step 3: 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

Key Concepts

Mass Calculation

When you are tasked with calculating the mass of a cylinder-shaped object like a penny, the starting point is to determine its volume. A penny can be represented as a cylinder. To find its volume, use the formula \[ \text{Volume} = \pi \times (\text{radius})^2 \times \text{height} \]After you find the volume, the next step is to calculate mass using the material's density. Density is like a bridge between volume and mass. It tells us how much mass is contained in a given volume. The formula to calculate the mass is:\[ \text{Mass} = \text{Density} \times \text{Volume} \]Since the density of copper is 8.9 \(\mathrm{g/cm^3}\), you multiply this value by the volume of the penny to find its mass. This approach helps in understanding how volume, density, and mass are interrelated.

Density of Copper

Density is an important concept that indicates how compact a material is. For copper, a very dense metal, it is given as 8.9 \(\mathrm{g/cm^3}\). This means each cubic centimeter of copper weighs 8.9 grams. This density is used to convert the volume of copper into mass.

• The given density allows you to understand how heavy a material will be for its size.
• Since copper is dense, it means even a small volume has a considerable weight.
• In real-world applications, knowing the density helps in estimating the mass when you only know the volume.

Being proficient with such calculations is beneficial for tasks involving metalwork or materials purchase for manufacturing purposes.

Conversion from Grams to Pounds

Sometimes, you need to express mass in different units, such as converting from grams to pounds. Since the metric and imperial systems are widely used, being able to convert between these systems is crucial.To convert grams to pounds, use the conversion factor:\[ 1 \text{ pound} = 453.592 \text{ grams} \]Therefore, you divide the total mass in grams by 453.592 to convert it into pounds:\[ \text{Mass (pounds)} = \frac{\text{Mass (grams)}}{453.592} \]Understanding how to switch between units can make it easier to grasp international measurements or communicate with different scientific audiences. It also facilitates comparing material prices sold by weight in different regions.

Metal Value Estimation

Estimating the value of a metal component involves understanding both its weight and current market price. Once you have mass in the appropriate unit (pounds in this example), you can multiply by the market price to get the value.

• Start by determining the metal's mass in pounds.
• Use the market price per pound to determine the total cost value.
• The formula is:\[ \text{Value} = \text{Mass (pounds)} \times \text{Price per pound} \]

In this specific case, the value of copper in a penny was calculated using the price of \(2.40 per pound, leading to an estimated copper value of approximately \)0.008. Being skilled at these estimations can assist in financial analyses, cost projections, or budgeting for materials.