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# Units — Harder example

## Video transcript

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 three point two eight feet so they're converting linear meters to linear fee they're not telling us how many that one cubic me they're not saying that well 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 cubic cubic meter so how would we think about this well and if you're taking 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 we're going to 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 1 meter in this dimension that's going to be three point two eight feet so maybe in us I'm not obviously drawing it perfectly but that's maybe a foot that's another foot actually I drove too small that's a foot that's a foot and that's a foot so you have one two three and then a little bit a little more than 1/4 of a foot so that's three point two eight feet and then let's see 1 foot 2 foot 3 foot maybe have one foot one foot 2 foot 3 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 could put 2.1 computers in this small cubic foot right over here surely you're going to be able 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 you can just you can just you can figure out how many cubic feet it is you can just multiply by the dimensions in terms of feet so this right here it's going to be three point two eight feet in that dimension it's going to be three point two eight feet high and it's going to be three 0.28 feet I guess we call this dimension maybe deep and so what's the volume right over here well it's going to be three point two eight times three point two eight times three point two eight so the volume here is three point two eight to the third power feet cubed cubic feet and this is how many cubic feet you have per cubic metre because this whole thing is also a cubic meter now so this is the number of cubic feet per cubic meter and if you want to know how many computers you can fit in that you can then just multiply by the density in cubic feet so times two point one 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 them you're going to get three point two eight to the third power times two point one that's this times this Computers Computers per meters cubed or per cubic meter so this is this is a 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 three point two eight 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 six point eight nine and it's completely reasonable these numbers look right that three point two eight to the third power times two point one actually would be around seventy four point one and so this is definitely 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 based on these choices you can actually just deduce that this is going to be the right choice this choice is interesting because if you look at it it's this is this is this is a 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 two point one computers per cubic foot and then if you just say yeah well I'll have three point two eight as many feet per meter not thinking in terms of cubic feet and cubic meters and if you take two point one times three point two eight that's probably this choice here I haven't multiplied it out but that looks like six point eight nine so that's where this answer actually came from