1.2.2 Multi-Step Dimensional Analysis Conversions

Continuing on some dimensional analysis problems, let’s take a look at these two. For problem number five, we want to change five hundred thousand seconds and see how many days that is. And then we also want to look at how many minutes there are in one week. In each of these problems, I do know lots of relationships about time, but to be honest, I don’t know how many seconds there are in a day or how many minutes there are in a week. You can sit and try to figure this out on your own, But to be honest, if we’re using this dimensional analysis process, we can let the process sort out our work and keep us organized to make sure we don’t miss anything. Let’s see how we can use the dimensional analysis process to our advantage. In this case, I’m going to start out just like before. For problem #5, I want to start with five hundred thousand seconds. Now because I’m going to be working with unit fractions, think of that or remember that that seconds is on top. You don’t have to write it over one, but you do need to make sure you think of it as being on top. What I’d like to do is I’d like to go from seconds to days, But I just don’t know off the top of my head how many seconds there are in a day. But I do know how many seconds there are in a minute, and so I’m just going to kind of start here. I know that in one minute there’s 60 seconds. Again, double check and make sure the 60 has to go with the seconds. 60 minutes is NOT equal to one second, so you’ve got to make sure the number goes with the word that has that equal value. So what I’ve got so far is the seconds cancel out. Now if I stop right now, what I’ll have is how many minutes five hundred thousand seconds is. But that’s not really what I want. I’d like how many days there are! I can multiply by conversion fractions or I can multiply by one as many times as I want without changing the problem. So I can actually multiply by a whole String of conversion fractions, and that’s what I’m going to do to solve this. Right now I’ve gotten the seconds to cancel out, and I’m to minutes. But I don’t really want minutes. What I want to get to is days. So I start here with minutes and I don’t know how many minutes there are in a day off the top of my head. But I do know how many minutes there are in an hour. There’s 60 minutes in one hour. Now the minute units are going to cancel and now what I have is I have set up a problem that’s going to change me from seconds into hours. Which is great! I’m lots closer, but not quite where I want to be. I’d still like to get to days. So let’s keep this trail up! Right now I have hours on the top and I don’t want hours. So I need hours on the bottom to cancel, and I’d like to change them to days. Now in this particular case, I know that there are 24 hours in one day, and I can fill those values in. So the great thing about this now is I’m almost done! I can multiply all the way across the top and multiply all the way across the bottom and see what’s going to happen. Because what’s left in terms of units is only days, and so as long as my numbers get resolved I’ll have my solution. So multiplying across the top: five hundred thousand times lots of one’s gets me five hundred thousand on the top. On the bottom I need to do 60 x 60 x 24, so let’s pull that up. 60 x 60 x 24 and this will give me 86,400. And then I can just do 500,000 divided by that 86,400, and that will be my solution. So 500,000 divided by 86,400 gets me — we’ll just round to two decimal places again 5.7…that will round up to nine. 5.79 and then my unit that I wanted – what’s left from my conversion – is days. So five hundred thousand seconds is the same as almost six days, that is 5.79 days. So I can use this concept of creating a whole trail of unit conversions. Then just make sure that any time that I have a conversion fraction, things are cancelling out and I’m moving closer and closer to my goal. So let’s do that same thing here. How many minutes are there in a week? Well that means I want to start with one week and I want to convert this until I get to where there’s minutes. Again, the week is kind of like it’s on top. If I’m working my way to minutes, I don’t know how many minutes there are in a week, but I do know how many days there are in a week! So week unit on top and week unit on the bottom will cancel out, and get me to days. And there’s seven days in one week. Okay, I don’t want days. So I keep going. If there are seven days in a week… So I want to get rid of days. Days are on the top, so they go on the bottom and I want to change. I don’t know how many minutes there are in a day, but I can go from days to how many hours there are in a day, and I know that there’s 24 hours in one day. That’ll get my days units to cancel and now I have changed weeks into hours. Hours aren’t quite what I want yet. I want minutes. So I’m going: hours are on top so I write hours on the bottom. That will get those to cancel out, and I would like to go to minutes, and I do in fact know how many minutes there are in an hour! There’s 60 minutes in one hour so I can fill those in. Now the hours cancel. I’m left just with minutes. Notice that every unit along the way has an equal measurement on top and bottom. Seven days in a week, 24 hours in a day, 60 minutes in an hour, and I can now get my solution from here. I’m going to multiply all the numbers across the top so 1 x 7 x 24 x 60. Let’s pull that up here. 1 x 7 x 24 x 60 and that gives me 10,080. Multiplying across the bottom is 1. Divided and we get 10,080 minutes every week. So this dimensional analysis is super, super, super helpful. We can do it if we know just the single conversion, but if we need multiple conversions we can use that to help us out. This concept of multiplying by a whole chain of conversion factors is really important when we need to convert multiple units — sometimes we call them compound units or rates — which is what we’ll do in the next video.

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