Question

A copper refinery produces a copper ingot weighing \(150 \mathrm{lb}\). If the copper is drawn into wire whose diameter is \(8.25 \mathrm{~mm}\), how many feet of copper can be obtained from the ingot? The density of copper is \(8.94 \mathrm{~g} / \mathrm{cm}^{3}\) (Assume that the wire is a cylinder whose volume is \(V=\pi r^{2} h\), where \(r\) is its radius and \(h\) is its height or length.)

Short answer

Approximately 1479.6 feet of copper wire can be obtained from the copper ingot.

Step-by-step solution

1. Convert the weight of the copper ingot from pounds to grams.

To convert the weight of the copper ingot from pounds (lb) to grams (g), we use the following conversion factor: \(1 \mathrm{~lb} = 453.592 \mathrm{~g}\). So, the weight of the copper ingot in grams is: \(150 \times 453.592\) \(= 68038.8 \mathrm{~g}\)

2. Calculate the volume of the copper ingot.

We can calculate the volume of the copper ingot by using the weight and the density of copper: \(Volume = \frac{Weight}{Density}\) \(Volume = \frac{68038.8 \mathrm{~g}}{8.94 \mathrm{~g/cm^3}}\) \(Volume = 7608 \mathrm{~cm^3}\)

3. Calculate the radius of the copper wire from the given diameter.

The diameter of the copper wire is given as \(8.25 \mathrm{~mm}\). To find the radius, we need to divide the diameter by 2 and also convert it to centimeters: \(Radius = \frac{8.25 \mathrm{~mm}}{2 \times 10} = \frac{8.25 \mathrm{~mm}}{20} = 0.4125 \mathrm{~cm}\)

4. Determine the volume of the copper wire as a cylinder.

We are given the formula for the volume of a cylinder: \(V = \pi r^2 h\). As we now have the value for the radius, we can rewrite the formula for the volume of the wire in terms of its length: \(7608 \mathrm{~cm^3} = \pi (0.4125)^2 h\)

5. Calculate the length of the copper wire obtained from the copper ingot.

Now we will solve the equation for the length (height) of the copper wire: \(h = \frac{7608 \mathrm{~cm^3}}{\pi (0.4125)^2}\) \(h = 45071.4 \mathrm{~cm}\) To convert the length from centimeters to feet, we use the conversion factor \(1 \mathrm{~ft} = 30.48 \mathrm{~cm}\): \(Length = \frac{45071.4 \mathrm{~cm}}{30.48 \mathrm{~cm/ft}} = 1479.6\mathrm{~ft}\) Therefore, approximately 1479.6 feet of copper wire can be obtained from the copper ingot.

Key concepts

Density

Understanding density is essential when dealing with material properties and calculations. Density is a measure of how much mass is contained in a given volume. It's expressed as mass per unit volume (like grams per cubic centimeter, \(g/cm^3\)). In our copper wire problem, the density of copper is given as \(8.94 g/cm^3\). This means for every cubic centimeter of copper, there is 8.94 grams of mass.

When you have a certain weight of the material, like the 150 lb copper ingot from our exercise, you can use the density to figure out the volume of the material. The concept of density allows us to transition from the weight of an object to its physical size, which is necessary for the rest of the calculations involving the dimensions of the copper wire.

Unit Conversion

Unit conversion is a key skill in science and engineering, as it allows us to express quantities in different units and compare them effectively. Conversion factors are used to transition from one unit to another. For example, in the provided copper wire exercise, we converted the mass of the copper from pounds to grams using the factor \(1 lb = 453.592 g\). Additionally, the length of the wire was ultimately needed in feet, so after calculating it in centimeters, we converted it using the factor \(1 ft = 30.48 cm\).

Tip for Success:

Always double-check that you convert units properly and keep track of them throughout calculations. Improper unit conversion can lead to incorrect results, despite accurate mathematical work.

Cylinder Volume

The volume of a cylinder plays a vital role in this exercise since the wire is assumed to have a cylindrical shape. The volume of a cylinder is found using the formula \(V = \pi r^2 h\), where \(r\) is the radius of the base and \(h\) is the height, or in this case, the length of the wire. To find the volume of any cylinder, you need to square the radius of the base, multiply it by \(\pi\), and then by the height of the cylinder.

In our copper wire example, we need to find the length of the wire. After calculating the volume that the mass of copper will occupy (using density), we rearrange the cylinder volume formula to solve for \(h\), the height or length of the wire. This is a practical application of geometry in real-world problem-solving.

Stoichiometry

Stoichiometry is a section of chemistry that involves calculating the amounts of reactants and products in chemical reactions. However, in a broader sense, it entails the quantitative relationship between the constituents of a substance or system. In our context, we apply a stoichiometric method by looking at the relationship between mass, density, and volume. The stoichiometry helps us understand that if we know two properties (mass and density), we can compute the third (volume).

In the copper wire problem, we carried out a stoichiometric calculation: we divided the mass of the copper ingot by the density of copper to find the volume it would occupy. Utilizing such relationships allows us to bridge the gap between the chemical properties of a substance and its physical geometry, such as calculating the length of wire produced from a metal ingot.