The masses and volumes of five objects are measured and the results are plotted on the shown graph. Which object has the highest density?
In this graph, we see the mass of these five objects are plotted on the vertical axis. And their volume is plotted on the horizontal. The five objects are labelled 𝐴, 𝐵, 𝐶, 𝐷, and 𝐸. And the data point, the dot, shows where their mass and volume points lie. So for example, for object 𝐴, the volume that that object takes up is a bit less than 1000 units of volume. And its mass looks to be about 1.1 mass units. And it’s the same idea with the other four objects. Each of them has a relative volume and mass as well.
Based on these data points, we want to figure out which object has the highest density of them all. We can recall that the density, 𝜌, of an object is equal to its mass, 𝑚, divided by the volume that it takes up, 𝑣. This means that in order to calculate object density, we’ll need to know the object’s mass and volume. To begin figuring out which of these five objects has the highest density, let’s consider them in terms of their mass and, separately, their volume.
So, looking at these objects through the perspective of their mass, we can create a ranked list which shows which objects have the most mass relative to the others. The way we can do that is start at the top of the vertical axis, and then just walk our way down and record each time we cross over one of the mass values of our object.
Starting from the top of the graph and working down, the first mass we encounter is mass 𝐵. So, that’s the first mass value we’ll record. We’ll call it 𝑚 sub 𝐵. Then, as we go a bit farther down the axis, soon we encounter the mass of object 𝐴. Since mass 𝐵 is greater than mass 𝐴, we’ll write 𝑚 sub 𝐵 is greater than 𝑚 sub 𝐴. Then, we continue on down the axis until we run into the mass value for object 𝐶. So, we can write that 𝑚 sub 𝐴 is greater than 𝑚 sub 𝐶. And then continuing on down, we arrive at object 𝐷. So, the mass value 𝑚 sub 𝐶 of object 𝐶 is greater than the mass of object 𝐷. And finally, we get to the mass value of object 𝐸. This is the smallest of all the five mass values. So we record that in our list, that 𝑚 sub 𝐷 is greater than 𝑚 sub 𝐸.
Looking at this ranked list of masses for our five objects, we see that because density is directly proportional to mass, that is, the more mass goes up, the more density goes up, we start to see that the objects towards the left of our list, 𝑚 sub 𝐵 and 𝑚 sub 𝐴, are more likely to indicate the highest density for these different objects. But we won’t know that for sure until we also include the volumes of these objects. So, let’s now do the same thing along the horizontal axis for the volumes of these five objects.
Starting out at the largest volume value of 2500, we then move in until we reach the volume of object 𝐶. We’ll call that volume 𝑣 Sub 𝐶. And we know it’s the greatest of the volumes of all five objects. Then, just a short bit later, we get to the volume of object 𝐷. So, we can say 𝑣 sub 𝐶 is greater than 𝑣 sub 𝐷. Continuing on, we encounter the volume of object 𝐵. So then, 𝑣 sub 𝐷 is greater than 𝑣 sub 𝐵, the volume of object 𝐵. And then, moving along down our axis, we reach the volume of object 𝐸. So we include that in our ranked list, 𝑣 sub 𝐸. And then lastly, we reach the volume of object 𝐴. It’s the smallest of all five object volumes. And so it comes at the end of our list, 𝑣 sub 𝐴.
Now, as we consider volume, we see that in the equation for density, volume is in the denominator, meaning that the smaller volume is, the higher density would be. So this means our smaller volumes, the ones towards the right-hand side of our list, are more likely to correspond to objects with higher density.
We’ve now looked at both mass and volume for each of these five objects. And it’s time to figure out which object has the greatest density. To figure this out, we’ll look at the left side of our ranked list for masses — because the higher the mass, the greater the density — and the right side of our list for volumes — because the lower the volume, the greater the density — and see if we find any overlap. What we mean by overlap is the mass of a given object and the volume of that same object appearing both to the left of the ranked mass list and to the right of the ranked volume list.
Keeping an eye out for that, we see that 𝑚 sub 𝐴, the mass of object 𝐴, is one of the leftmost masses, therefore one of the greatest masses, and that 𝑣 sub 𝐴, the volume of object 𝐴, is literally the smallest volume. This means that if we were to calculate the density of object 𝐴, we would use the second largest mass of them all. And we would use the very smallest volume of them all. This combination, 𝑚 sub 𝐴 divided by 𝑣 sub 𝐴, would yeild to the largest density of any of these five objects. And that’s our answer. We’ll say that object 𝐴, based on its mass and volume relative to the other four objects, has the highest density.