Volume density

In this video, we're going to talk a little bit about density, and we're especially going to talk about density in the context of volume. One simple way to think about density is it's a quantity of something, and we're going to think about examples of it per unit volume, so per volume.

For example, let's say that I have a cubic meter right over here. Actually, let me have two different cubic meters just to give you an example. So these are both cubic meters, and let's say in the one on the left I have a quantity of, let's say, six of these dots per cubic meter. And over here, I only have three of these dots per cubic meter. Well, here I have a higher density.

In general, we're going to take the quantity and divide it by the volume, and the units are going to be some quantity per unit volume. Now, you're typically going to see mass per unit volume, but density, especially in the volume context, can refer to any quantity per unit volume.

Now, with that out of the way, let's give ourselves a little bit of an example. So, here we're told that stone spheres thought to be carved by the Decay people—I'm not sure if I'm pronouncing that correctly—more than a thousand years ago are a national symbol of Costa Rica. One such sphere has a diameter of about 1.8 meters and a mass of about 8,300 kilograms.

Based on these measurements, what is the density of this sphere in kilograms per cubic meter? Round to the nearest hundred kilograms per cubic meter. So pause this video and see if you can figure that out.

All right, so we're going to want to get kilograms per cubic meter. We know the total number of kilograms in one point in a sphere that has a diameter of 1.8 meters, so that's the total number of kilograms, but we don't know the volume just yet.

So we have a sphere like this. This would be a cross-section of it; its diameter is 1.8 meters. Now, you may or may not already know that the volume of a sphere is given by four-thirds pi r cubed, and so the radius here is 0.9 meters, and so that would be the r right over here.

The volume of one of these spheres is going to be—let me write it over here—the volume is going to be four-thirds pi times 0.9 to the third power. And we know what the mass is. The mass in that volume is 8,300 kilograms, so we would know that the density, the density in this situation is going to be 8,300 kilograms per this many cubic meters: four-thirds pi times 0.9 to the third power cubic meters.

And we're going to need a calculator for this, and we're going to round to the nearest hundred kilograms. So we have 8,300 divided by—let me just open parentheses here—four divided by three times pi times 0.9 to the third power. And then I'm going to close my parentheses.

It is equal to this right over here. We want to round to the nearest hundred kilograms, so approximately 2,700 kilograms per cubic meter. 2,700 or 2,700 kilograms per cubic meter, and we are done.