A copper refinery produces a copper ingot weighing 70 kg. If the copper is drawn into wire whos

A copper refinery produces a copper ingot weighing 70 kg....

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.)

Key Concepts

Mass-Volume Relationship

The relationship between mass and volume, often mediated by the concept of density, allows for the determination of one quantity given the others. This concept is fundamental in scenarios where the material’s mass is known and one needs to find its volume, which can then be used in further calculations, such as determining the length of material in a given shape.

Unit Conversion

Unit conversion is a key concept that involves converting measurements from one unit system to another to ensure consistency in calculations. In problems involving dimensions and physical properties, converting units such as kilograms to grams or millimeters to centimeters is vital for obtaining correct and comparable results.

Volume Calculation for Cylinders

The volume of a cylinder can be calculated using the formula V = ?r²h, where r is the radius of the circular base and h is the height (or length) of the cylinder. This formula is essential in problems where objects such as wires or rods are modeled as cylindrical shapes, allowing one to relate geometric dimensions to volume.

Density

Density is a fundamental physical property defined as mass per unit volume. It is used to relate the mass of a substance to the space it occupies, which is crucial when converting a known mass into volume or vice versa in various practical applications.

Transcript

00:01 If we're given a copper ingot and we know that it's going to make, or we know that it weighs 70 kilograms, we want to know how many meters of copper wire it can be drawn into.

00:13 Okay.

00:15 So the wire is going to be in the shape of a cylinder.

00:17 So for the volume, we'll use pi r squared times height.

00:20 We have the radius because we have the diameter of the wire.

00:27 So our radius from here to here will be half of our diameter.

00:32 So our radius then is 3 .75 millimeters, but we want this in centimeters because our density is in centimeters.

00:45 So just to keep our units straight, so we get point, divide by 10, 0 .375 centimeters for our radius to our radius.

01:00 Now we can plug this in for our volume value, and we have our mass, okay, and we have our density.

01:10 So really we're solving for the height.

01:12 We want to know how we're solving for here is our height, and we want to find the height, because it'll show us how much wire we can make, okay? so we'll set this up as 8 .94 grams per centimeter or cubic centimeter times, oh, is equal to, because we're setting this up, is an equation already, is equal to our mass.

01:53 Oh, that's in kilograms, so we want it in grams, so we'll divide by 1 ,000.

01:58 So our mass will be 0 .07 grams over our volume, which is pi times 0 .375 squared times our height.

02:24 I just realized i do this all the time...

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