### Video Transcript

Two spheres have the same density, but the first sphere has a volume 10 times greater than that of the second. How much greater is the mass of the first sphere than that of the second sphere?

Okay, so in this question we have two spheres. Let’s represent these spheres by drawing two circles of different colors. Let’s say that our orange circle represents the first sphere, so we’ll label it with a one, and our pink circle represents our second sphere, so we’ll label it with a two. Notice that we’ve drawn these circles different sizes, and this is because we’re told in the question that the volumes of the spheres they represent are different. Specifically, we’re told the volume of the first sphere is 10 times greater than the volume of the second sphere. So let’s call the volume of the first sphere 𝑉 one, and let’s call the volume of the second sphere 𝑉 two.

We know that 𝑉 one is 10 times greater than 𝑉 two, and we can write this as an equation as 𝑉 one equals 10 times 𝑉 two. Let’s rewrite this equation under our spheres so that we can have some space for the rest of the question.

Okay, we’re also told that the two spheres have the same density. So if we denote the density of the first sphere with the symbol 𝜌 subscript one and we denote the density of the second sphere with the symbol 𝜌 subscript two, we know these two things are equal, which means that 𝜌 one is equal to 𝜌 two. This is just an equation that tells us the two spheres have the same density. In fact, since these densities are the same, we don’t need to keep writing these subscripts, and we can just say that both spheres have density equal to 𝜌.

Given all this information, we’re asked to find the difference in mass between the two spheres. We can do this by first recalling the general formula for the density of an object, which is that the density of an object 𝜌 is equal to the mass of that object 𝑀 divided by the volume of that object 𝑉.

So let’s take a look at this equation for each of our two spheres. For sphere one, we have the density, which is 𝜌, is equal to the mass of sphere one, which we can call 𝑀 one, divided by the volume of sphere one, which is 𝑉 one. And for sphere two, we have the density, which is also 𝜌, is equal to the mass of sphere two, which we can call 𝑀 two, divided by the volume of sphere two, which is 𝑉 two. So we now have two equations, the first of which came from sphere one and the second of which came from sphere two.

But let’s remember that the densities of our two spheres are the same, so 𝜌 here is equal to 𝜌 here. This means that the right-hand sides of our two equations, 𝑀 one over 𝑉 one and 𝑀 two other 𝑉 two, must also have the same value. And this means we can simply write them equal to each other. So we have 𝑀 one over 𝑉 one is equal to 𝑀 two divided by 𝑉 two. And this is true because both spheres have the same density.

Now the question asked us specifically about the mass of the first sphere compared to the mass of the second sphere, so let’s make the mass of the first sphere the subject of this equation. We can do this by multiplying both sides of the equation by 𝑉 one. And on the left-hand side of the equation, we can cancel the 𝑉 one on the top of the fraction with the 𝑉 one on the bottom of the fraction, meaning the left-hand side just reads 𝑀 one. Meanwhile, the right-hand side reads 𝑀 two divided by 𝑉 two times 𝑉 one, which we can write as 𝑉 one times 𝑀 two divided by 𝑉 two. This is helpful because we can recall that we know how 𝑉 one and 𝑉 two are related. Specifically, we know that 𝑉 one is equal to 10 times 𝑉 two.

So in our equation for 𝑀 one, we can replace 𝑉 one with 10 times 𝑉 two. This leaves us with an equation that reads 𝑀 one is equal to 10 times 𝑉 two times 𝑀 two divided by 𝑉 two. This is good because we now have a 𝑉 two on the top of the fraction and on the bottom of the fraction on the right-hand side. So we can cancel out these two factors of 𝑉 two, meaning the right-hand side of the equation simply reads 10 times 𝑀 two. And the whole equation tells us that the mass of the first sphere 𝑀 one is equal to 10 times the mass of the second sphere 𝑀 two.

And this actually gives us our final answer for this question since we were asked to find out how much greater the mass of the first sphere is compared to the mass of the second sphere. So we can give our final answer as 10. The mass of the first sphere is 10 times greater than that of the second sphere.