Units — Harder example

- [Instructor] In order to connect to the internet, dedicated computers are kept in a server room. To prevent overheating, the density of computers in a server room must not exceed 2.1 computers per cubic foot. One meter is about 3.28 feet. So they're converting linear meters to linear feet. They're not telling us how many that one cubic meter. They're not saying how many cubic feet is one cubic meter, so I suspect we're gonna have to figure that out. Which of the following densities is equal to the maximum number of computers per computers per cubic meter? So how would we think about this? Well, and if you're taking the SAT, you might not have time to draw a nice diagram, but this is literally what I am visualizing in my head when I first read that problem is I imagine a cubic meter. I imagine a cubic meter. And along each of the three dimensions in the space we know of, the three-dimensional space, a cubic meter, it's going to be one meter in this dimension by one meter in this dimension by one meter in that dimension. That's what a cubic meter is. Now, one meter in this dimension, that's going to be 3.28 feet. So maybe, and this, I'm not obviously drawing it perfectly, but that's maybe a foot. That's another foot. Actually, I drew 'em too small. That's a foot, that's a foot, and that's a foot. So we have one, two, three, and then a little bit, a little more than 1/4 of a foot. So that's 3.28 feet. And then let's see, one foot, two foot, three foot. Then you have one foot, one foot, two foot, and three foot. And so a cubic foot will actually look something like this. A cubic foot would look something like this. So if you could put 2.1 computers per cubic foot, if you can put 2.1 computers in this small cubic foot right over here, surely you're going to be able to put a lot more than 2.1 computers in the cubic meter. And just off of that, you can rule out these two choices because these two are less than 2.1. But how would you actually calculate it? Well, you could just multiply this volume, a cubic meter. To figure out how many cubic feet it is, you can just multiply by the dimensions in terms of feet. So this right here is going to be 3.28 feet in that dimension. It's going to be 3.28 feet high. And it's going to be 3.28 feet, I guess we can call this dimension maybe deep. And so what's the volume right over here? Well, it's going to be 3.28 times 3.28 times 3.28. So the volume here is 3.28 to the third power feet cubed. Cubic feet. And this is how many cubic feet you have per cubic meter because this whole thing is also a cubic meter. So this is the number of cubic feet per cubic meter. And if you wanna know how many computers you can fit in that, you can then just multiply by the density in cubic feet. So times 2.1, I'll write comp for computers, computers per cubic feet. And you can see that these units are going to work out. You have cubic feet divided by cubic feet. And if you multiply 'em, you're going to get 3.28 to the third power times 2.1, that's this times this, computers per meters cubed or per cubic meter. So this is the maximum number of computers per cubic meter right over here. Now, we could try to solve this by hand or solve it some way, but we can estimate it because these two remaining choices are quite different. What's 3.28 to the third power going to be roughly? Well, three to the third power is 27, so this thing is going to be larger than 27. And if you multiply something larger than 27 by something larger than two, you're going to get something larger than 54. So this thing is definitely going to be larger, way larger than 6.89. And it's completely reasonable. These numbers look right. 3.28 to the third power times 2.1 actually would be around 74.1. And so this is definitely the choice that I would pick. Now, if you had a lot of time, you could multiply this out, use your calculator, but basically these choices, you could actually just deduce that this is going to be the right choice. This choice is interesting, 'cause if you look at it, this is kind of an answer to pick if you forgot to convert to cubic, if you forgot to convert how many cubic feet are per cubic meter and if you just kept things in linear meters and feet, because then you might say, "Okay, if I have 2.1 computers per cubic foot," and then if you just say, "Hey, well, I'll have 3.28 as many feet per meter," not thinking in terms of cubic feet and cubic meters. But if you take 2.1 times 3.28, that's probably this choice here. I haven't multiplied it out, but that looks like 6.89. So that's where this answer actually came from.