Linear equations word problem: marbles
About this video
Andrei fills a glass tank with marbles and then water, and the water needed is W(n) = 32 - 0.05n liters. Sal shows that W(0) = 32 gives the tank's volume, and that each extra marble reduces the water by 0.05 liters, which is the volume of one marble.
You'll be able to
- interpret the starting value of a linear function in context.
- interpret the rate of change of a linear function in context.
- use a function like W(n) = 32 - 0.05n to answer questions about a situation.
Key words
- function
- A rule that gives one output for each input, like W(n) giving liters of water for n marbles.
- W(0)
- The output of the function when the input is zero; here, the water needed with no marbles.
- volume
- How much space something takes up, measured here in liters.
Transcript
- Andrei has a 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. This is the formula for the volume of water Andrei should use W, measured in liters, as a function of the number of marbles used n.
And we see that the amount of water he needs to put in as a function of the marbles used, n, is equal to 32 minus 0.05n. That's interesting. Complete the following sentences with the appropriate missing values. The glass tank's volume is blank liters. The volume of each marble is blank liters.
So, one thing that we could do to think about the glass tank's volume is, is well, let's use this function to figure out how much water would we need if we had no marbles in it, because that amount of water would be the volume of the entire tank. Remember, we're gonna fill up the entire tank with water based on how many marbles are in it, but if we figure out what W of zero is, this is the amount of water we need if we had no marbles.
This would be the volume of the entire tank and what is this going to be? Well W of zero, when n is equal to zero, this term goes away and we're left with 32. So, W of zero would be volume of entire tank. Volume of tank, which is equal to 32 liters and so, we could write 32 right over there.
And then they ask us, the volume of each marble is blank liters. Well, what's going on here in this function definition? So, the highest that this could be is 32, 'cause that's the most water you would put in. If you had no marbles, you would just fill up the tank and put in 32 liters.
Then, every marble that you put in, you're going to have to put in 0.05 less liters. You see that, you have minus 0.05 times n. So, every time n grows by one, you have to put in 0.05 less fewer liters of water, which implies that the marble is 0.05 liters. That's it's volume, so let's write that down, 0.05 liters.
And, you could test this assumption. If you said, look, if you had one marble, if you had one marble, then you would have to put 0.05 less than 32 liters in and why would that be the case? Well, that would mean because that one marble is taking up 0.05 liters, that's how much space it's taking up.
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