Question

A copper refinery produces a copper ingot weighing $70\mathrm{kg}$. If the copper is drawn into wire whose diameter is $7.50\mathrm{mm}$, how many meters 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 $r$ is its radius and $h$ is its height or length.)

Short answer

Approximately 210 meters of copper wire can be obtained from the ingot.

Step-by-step solution

Find the volume of the copper ingot

We are given the mass of the copper ingot and the density of the copper. We can use the following formula to find the volume: Volume = Mass/Density Mass of ingot = 70 kg Since we will later need the density and mass to have matching units, let's first convert mass from kg to g: Mass of ingot = $70\times 1000=70,000\mathrm{g}$ Density of copper = 8.94 g/cm³ Volume of ingot = $\frac{70,000\mathrm{g}}{8.94{\mathrm{g}/\mathrm{cm}}^3}$ Now, calculate the volume of the ingot: Volume of ingot = $7834.45{\mathrm{cm}}^3$

Find the radius of the copper wire

We are given the diameter of the copper wire which is 7.50 mm. To find the radius, we just need to divide the diameter by 2: Radius = Diameter/2 Radius = $7.50\mathrm{mm}/2$ Now, convert the radius from mm to cm as we will later need the radius and volume to have matching units: Radius = $0.375\mathrm{cm}$

Use the volume formula for a cylinder to find the length (height) of wire

The volume formula for a cylinder is: $$V=\pi r^{2}h$$ Since we are looking for the height (length) of the wire, rearrange the formula for h: $$h=\frac{V}{\pi r^2}$$ We've already found the volume of the ingot and the radius of the wire, so we can plug in these values and calculate the height (length) of the wire: $$h=\frac{7834.45{\mathrm{cm}}^3}{\pi (0.375\mathrm{cm}{)}^2}$$ Now, calculate the length (height) of the wire: Length of wire = 21062.25 cm Finally, convert the length from cm to meters: Length of wire = $210.62\mathrm{m}$ So, approximately 210 meters of copper wire can be obtained from the ingot.

Key concepts

Volume of Cylinder

When discussing the volume of a cylinder, imagine a 3D shape with two circular ends and straight sides, much like a tin can. To find the volume, you need to calculate the space inside this shape. The formula used is:

• Volume = Area of Base × Height

The base is a circle, so to find the area of the base, we use the formula for the area of a circle, which is $A=\pi r^2$, where $r$ is the radius of the circle. Once you have that, multiply by the height, $h$, to obtain the volume:

• $V=\pi r^{2}h$

This formula helps to determine how much space the cylinder occupies. Understanding how to calculate volume is crucial when you're dealing with physical objects and need to figure out how much material is needed or what capacity a cylindrical container might have.

Density of Copper

Density is a key property of materials that helps us understand how much mass is contained within a certain volume. It is given by:

• Density = Mass / Volume

In this context, the density of copper is 8.94 g/cm³, which tells us that every cubic centimeter of copper has a mass of 8.94 grams. Knowing the density allows us to relate the mass of an object to the volume it occupies.

To find the volume using density, you need to rearrange the formula to solve for volume:

• Volume = Mass / Density

This relationship is particularly useful when working with metals like copper, where the mass and shape can change during processing, but the density remains consistent.

Unit Conversion

Unit conversion is the process of converting a measurement from one unit to another, ensuring that calculations are compatible across contexts. In this scenario, it involves changing mass from kilograms to grams, and measurements of length from millimeters to centimeters. Here's a brief overview of how to conduct these conversions:

• 1 kilogram = 1000 grams
• 1 millimeter = 0.1 centimeters

Effective unit conversion is crucial in scientific calculations to maintain consistency and avoid errors. When performing any calculation involving different units, convert all measurements to common units before applying formulas. This ensures accurate results and simplifies the calculation process.

Cylinder Volume Formula

The cylinder volume formula $V=\pi r^{2}h$ plays an essential role in solving problems where the cylinder is involved. It's derived from the circle area formula, adapted for a three-dimensional shape. This formula requires two main inputs:

• The radius ( of the cylinder's base
• The height ( , which could also be referred to as the cylinder's length in some contexts

Using this formula helps determine the amount of space within a cylindrical object. In practical terms, if you're calculating the length of a wire, you set the known volume (like from a copper ingot) equal to the cylinder volume formula and solve for height (or length), since the volume is a function of radius squared and height. Understanding how to manipulate this formula is invaluable for problems involving cylindrical shapes, including those in engineering and manufacturing.