A copper refinery produces a copper ingot weighing 150 \(\mathrm{lb}\) . If the copper is drawn into wire whose diameter is 7.50 \(\mathrm{mm}\) , 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 \(V=\pi r^{2} h,\) where ris its radius and \(h\) is its height or length.)

Density is a fundamental concept in physics and materials science, often denoted by the symbol \rho. It refers to the mass per unit volume of a substance and is an intrinsic property that doesn't change regardless of the amount of material. The formula used to calculate density is:

\[\begin{equation}\text{Density (}\rho\text{)} = \frac{\text{Mass (m)}}{\text{Volume (V)}}\end{equation}\]
In the context of our copper wire problem, understanding density is crucial. Copper has a density of 8.94 g/cm³. This constant allows us to calculate the volume of copper material when we know its mass, or vice versa. For a copper refinery, this is an everyday calculation to figure out how much product can be made from a certain mass of raw copper.

The volume of a cylinder is a measure of how much space it occupies, which is vital for tasks such as determining how much material is needed to make an object or how much product can be obtained from a specific amount of material. For a cylinder, we calculate volume as:

\[\begin{equation}V = \pi r^{2} h\end{equation}\]
where \(V\) is the volume, \(r\) is the radius of the circular base, \(h\) is the height or length of the cylinder, and \(\pi\) roughly equals 3.14159. When dealing with copper wire, which is a long, thin cylinder, calculating the volume allows you to determine the length of wire that can be produced from a chunk of copper. Understanding cylinder volume is a key concept for manufacturers dealing with rod- or tube-shaped materials.

Unit conversion is a process that allows us to convert measurements from one set of units to another, ensuring that calculations and comparisons are correct and meaningful. This is essential in different fields, especially in science, engineering, and everyday life.

When we work on problems like our copper wire exercise, converting units is critical for accuracy since values are given in different systems (e.g., pounds to grams, millimeters to centimeters, feet to centimeters). In this exercise, we used the conversion factors 1 lb = 453.592 g and 1 ft = 30.48 cm. It's important for students to be familiar with these conversion factors and how to apply them properly to solve problems efficiently and correctly. Constant practice with unit conversion will make it second nature, an essential skill for anyone working in fields that require precise measurements.