Question

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.)

Short answer

Weight of copper ingot in grams = 150 lb * 453.592 g/lb = 68038.8 g Volume of copper ingot = (Weight of copper ingot in grams) / (Density of copper) = 68038.8 g / 8.94 g/cm³ = 7610.23 cm³ wire_radius = (7.50 mm) / 2 * (1 cm / 10 mm) = 0.375 cm length = 1 ft * 30.48 cm/ft = 30.48 cm Volume of unit length = π * r² * length = π * (0.375 cm)² * 30.48 cm ≈ 13.292 cm³ Length of copper wire = (Volume of copper ingot) / (Volume of unit length of wire) = 7610.23 cm³ / 13.292 cm³ ≈ 572.36 ft

Step-by-step solution

Convert the weight of the copper ingot into grams

To find the volume of the copper ingot, we first need to convert its weight from pounds (lb) to grams (g) using the conversion factor 1 lb = 453.592 g. So: Weight of copper ingot in grams = 150 lb * 453.592 g/lb

Calculate the volume of the copper ingot using its given density

Given that the density of copper is 8.94 g/cm³, we can calculate the volume of the ingot using the formula: Volume = weight / density Volume of copper ingot = (Weight of copper ingot in grams) / (Density of copper)

Find the volume of a unit length of the cylindrical copper wire based on its diameter

The wire is a cylinder with diameter 7.50 mm and radius (r) of half the diameter. Converting the diameter to centimeters, we can find the radius: wire_radius = (7.50 mm) / 2 * (1 cm / 10 mm) The volume of a unit length (1 ft) of the cylindrical copper wire can be calculated using the formula: Volume of unit length = π * r² * length But first, convert the length from feet to cm (1 ft = 30.48 cm): length = 1 ft * 30.48 cm/ft Now, we can calculate the volume of 1 ft long copper wire: Volume of unit length = π * r² * length

Divide the volume of the copper ingot by the volume of unit length of the wire to find the total length of the copper wire

To find the total length of copper wire that can be obtained from the copper ingot, divide the volume of the copper ingot by the volume of the unit length of the wire: Length of copper wire = (Volume of copper ingot) / (Volume of unit length of wire) Once the necessary calculations are made, you will obtain the length of the copper wire that can be obtained from the 150 lb ingot.

Key concepts

Density

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.

Cylinder Volume Calculation

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

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.