# Lesson Video: Density Mathematics

In this video, we will learn how to calculate the density of an object given its mass and volume and convert between different density units.

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### Video Transcript

In this video, we will learn how to calculate the density of an object given its mass and volume and also convert between different units of density. We will begin by explaining what we mean by density and recall the formula we can use to calculate it.

Density is a way of comparing how heavy a material is for its size. It is a measurement of the amount of a substance contained in a certain volume. The density of any solid is the mass of the object divided by its volume. This leads us to the general formula density is equal to mass divided by volume.

The mass of an object is usually measured in kilograms or grams. The volume is measured in cubic meters or cubic centimeters. This leads us to two standard units for density, either kilograms per cubic meter or grams per cubic centimeter. The general formula density is equal to mass divided by volume is sometimes represented in the triangle shown. This means that we can also calculate the mass by multiplying the density by the volume or calculate the volume by dividing the mass by the density.

We will now use this formula to calculate the density of an object given its mass and volume.

True or false: A cylinder with a volume of one sixtieth of a cubic meter and a mass of 150 kilograms has a density of 9,000 kilograms per cubic meter.

We recall that the density of any object is equal to its mass divided by its volume. If the mass is measured in kilograms and the volume in cubic meters, then the units for density will be kilograms per cubic meter. In this question, we are told the cylinder has a mass of 150 kilograms and a volume of one sixtieth of a cubic meter. The density will therefore be equal to 150 divided by one sixtieth. Dividing by a fraction is the same as multiplying by the reciprocal of the fraction. This is sometimes known as KCF. We keep the first number the same, we change the sign to a multiplication, and we flip the fraction.

60 divided by one is equal to 60. So the density is equal to 150 multiplied by 60. 15 multiplied by six is equal to 90. This means that 150 multiplied by 60 is equal to 9,000. The density of the cylinder is 9,000 kilograms per cubic meter. As this is the value we were given in the statement, the correct answer is true. A cylinder with a volume of one sixtieth of a cubic meter and a mass of 150 kilograms will have a density of 9,000 kilograms per cubic meter.

In our next question, we will calculate the density of a cube.

A cube has a side length of 0.15 meters. If the mass of the cube is three kilograms, what is its density? Give your answer approximated to one decimal place if required.

Let’s begin by considering a cube which has side length 0.15 meters. The volume of any cube can be calculated by cubing the side length. In this question, we need to cube 0.15. This is equal to 0.003375. We also know that if the side length is in meters, the volume will be in cubic meters.

We have been asked to calculate the density of the cube. And we know that density is equal to mass divided by volume. As the mass is equal to three kilograms, the density will be equal to three divided by 0.003375. Typing this into the calculator gives us an answer of 888.8 recurring. We need to round this answer to one decimal place. The density of the cube is therefore equal to 888.9.

As our units for mass were kilograms and the units for the volume of the cube were cubic meters, then the units for density will be kilograms per cubic meter. The density of a cube with side length 0.15 meters and a mass of three kilograms is 888.9 kilograms per cubic meter.

In our next question, we will calculate the mass of a ball given its density and radius.

A steel ball has a radius of 10 centimeters. If the density of the ball is 8,000 kilograms per cubic meter, find the mass of the ball in kilograms. Give your answer to one decimal place.

We are told that we have a ball with radius 10 centimeters. The volume of any ball or sphere is equal to four-thirds 𝜋𝑟 cubed. This means that we could calculate the volume of the ball in cubic centimeters by substituting 𝑟 equals 10 into the formula. As our units for density were kilograms per cubic meter, we actually need to convert the radius into meters first. As there are 100 centimeters in one meter, 10 centimeters will be equal to 0.1 meters.

We can therefore calculate the volume of the ball in cubic meters by multiplying four-thirds by 𝜋 by 0.1 cubed. Typing this into the calculator gives us an answer of 0.004188 and so on. We were asked to calculate the mass of the ball, which means this isn’t the final answer. We will therefore not round at this stage.

We know that the density of any object is equal to its mass divided by its volume. We can rearrange this formula so that the mass is equal to the density multiplied by the volume. The mass of the ball will be equal to 8,000 multiplied by 0.004188 and so on. This is equal to 33.5103 and so on. Rounding to one decimal place, we get an answer of 33.5. A steel ball with a radius of 10 centimeters, or 0.1 meters, and a density of 8,000 kilograms per cubic meter has a mass of 33.5 kilograms.

In our next question, we will need to calculate the volume given the density and mass of a block of aluminium.

The density of aluminium is 8,000 kilograms per cubic meter. Find the volume of a 100-kilogram block of aluminium.

We know that the density of any object is equal to its mass divided by its volume. The link between density, mass, and volume can also be demonstrated in the triangle shown. In order to calculate the volume of an object, we divide its mass by its density. If the mass is measured in kilograms and the density in kilograms per cubic meter, then the units for volume will be cubic meters.

In this question, the mass of the block of aluminium is 100 kilograms and the density is 8,000 kilograms per meter cubed. We can divide the numerator and denominator by 100. This means that the volume of our 100-kilogram block of aluminium is one eightieth of a cubic meter.

In our last question, we will look at how we can convert between different units of density.

The density of gold is 19,320 kilograms per cubic meter. What is this value in grams per cubic centimeter?

In order to answer this question, we need to work out how we can convert from kilograms per cubic meter to grams per cubic centimeter. We know that there are 1,000 grams in one kilogram. This means that in order to convert from kilograms to grams, we need to multiply by 1,000. There are 100 centimeters in one meter. In this question, we are dealing with cubic meters, the units of volume. Cubing 100 gives us one million. Therefore, there are one million cubic centimeters in one cubic meter. This means that to convert from cubic meters to cubic centimeters, we multiply by one million.

We can use this information to work out how we would convert from kilograms per meter cubed to grams per centimeter cubed. We need to multiply the numerator by 1,000 and the denominator by one million. Let’s consider the fraction 1,000 over one million. We can divide the numerator and denominator by 1,000, which gives us one over 1,000. This means that to convert from kilograms per meter cubed to grams per centimeter cubed, we need to multiply by one one thousandth. This is the same as dividing by 1,000.

We need to divide 19,320 by 1,000. Dividing by 1,000 moves all of our digits three places to the right. This means that the density of 19,320 kilograms per cubic meter is the same as 19.32 grams per cubic centimeter. The density of gold in grams per cubic centimeter is 19.32.

To convert between the two different units of density, we can divide or multiply by 1,000. To go from kilograms per cubic meter to grams per cubic centimeter, we divide by 1,000. And to go the other way, we multiply by 1,000.

We will now summarize the key points from this video. We found out in this video that the density of an object is equal to its mass divided by its volume. We also saw that we can rearrange the formula density equals mass divided by volume to help us calculate either the mass or the volume. Another way of demonstrating this is using the triangle shown. The mass of an object is equal to its density multiplied by its volume. And the volume is equal to the mass divided by the density.

We saw that the standard units of density are kilograms per cubic meter or grams per cubic centimeter. The units that the mass and volume are measured in will dictate which units we’ll use for density.

In the final question, we saw that we can convert between these two units of density by multiplying or dividing by 1,000. To convert from kilograms per cubic meter to grams per cubic centimeter, we divide by 1,000. And to convert from grams per cubic centimeter to kilograms per cubic meter, we multiply by 1,000.