Linear models word problem: marbles

- [Voiceover] William has a 26-liter glass tank. First, he wants to put some marbles in it, all of the same volume. Then, he wants to fill the tank with water until it's completely full. If he uses 85 marbles, he will have to add 20.9 liters of water. What is the volume of each marble? All right, so let's think about it. The volume of the marbles plus the volume of the water are going to be equal to the volume of the tank. They're going to fill up the tank. Let me write that down. So the volume, let me write V, V and I'll write a little subscript M here. This is the volume of the marbles plus the volume of the water, plus the volume of the water, are going to be equal to the volume of the tank. Are going to be equal to the volume of the tank. Now, we don't know what the volume of 85 marbles is, but we know, this right over here, this is going to be 85 times the volume of one marble. So this is volume of all the marbles, and maybe, I'll just write it out, volume, volume of one marble, one marble. I'll write it out. I could've put a variable in there, but just to make it clear. So 85 times the volume of one marble, that's gonna be the total volume of the marbles, plus the volume of water. They tell us what the volume of water is. 20.9 liters of water, plus 20.9 is gonna be equal to the volume of the entire tank. It's a 26-liter glass tank, so it's gonna be equal to 26. Remember, the whole thing that's going on here is he wants to fill up the whole tank. He puts some marbles in it that have some volume, and then whatever's left, he fills it with water. So these two volumes combined have to add up to the volume of the entire tank. Now, what is the, now we can just solve for the volume of one marble. And actually maybe I'll just, let's just call this the variable M. Let's just call this, let's call it M for the volume of one marble. So we get 85, we get 85 times M, maybe I'll just do this in one color for the sake of time, plus 20.9 is equal to 26. Now, to solve for M we can subtract 20.9 from both sides. So subtract 20.9, subtract 20.9 from both sides, and we get 85 times the volume of one marble is equal to, let's see, 26 minus 21 would have been five, so this is gonna be five point... Right, this is gonna be 5.1, It's gonna be five, 5.1, or we can say the volume of each marble if we divide both sides by 85 is 5.1 liters over 85. And I could calculate, let's see, let me just get a calculator out for this. We're allowed to use calculators on this one. And so we get 5.1 divided by 85, is equal to 0.06 liters, is equal to 0.06. So what is the volume of each marble? We just figured out 0.06 liters. And then how much water is necessary if William uses 200 marbles? So let's think about it. How much water would be necessary? Let's see, the volume, let's just say W is the amount of water that's necessary. And it's gonna be a function of the number of marbles, the number of marbles, and let's see, if you have, if you have no marbles you're going to have to put 26 liters, you have to fill the whole glass tank, the whole volume. So it's going to be equal to 26, minus the volume of the marbles. Well, what's the volume of the marbles going to be? Well it's going to be the volume of each marble. It's going to be 0.06 times the number of marbles you have, times the number of marbles you have. Just like that we've described a linear function that tells us how much water we have to fill as a function of marbles. It's gonna be the total volume of the tank minus the volume of the marbles, and the volume of the marbles is the volume per marble times the number of marbles. So if they're saying how much water we need for 200 marbles, that's going to be W, the amount of water we have to fill as a function of marbles, where now the number of marbles is 200. That's going to be equal to 26 minus 0.06 times 200, times 200. And what's that going to be? Well let's see, six times, this is six hundredths. Six hundredths times a hundred would be six, but now we're multiplying 200, so this is gonna be 12, so this thing right over here, this thing right over here is going to be equal to 12. And then 26 minus 12 is 14. So all of this, we need to write it like this, all of this is going to be equal to 14. So how much water is necessary if William uses 200 marbles? 14 liters of water. 26 minus 0.06 times 200.