Dimensional Analysis

by Jules Bruno
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so dementia analysis is seen as a failed proof process that allows you to convert from one unit to another. You designed the problems to begin with your given amount and to finish with the end amount of your unknown. Now just follow units to ensure the unwanted units are canceled out in the same way that we did metric prefix conversions where we lined things up on opposite levels to help them cancel out. We're gonna take that and apply it here to the mental analysis. Now the strategy is many. Conversion problems will utilize your given amount and conversion factors, and you're gonna use those together to help isolate your end amount. So if we take a look here in this general format, let's say we're given 32 inches so 32 inches air given amount, and they're asking us to identify how many centimeters this equals. So Centimeters is our end about what we're looking for. The conversion factor is a way of bridging these two ideas together, it's used to convert, are given amount into our end amount. Now to go from interest of centimeters means that we have to use a conversion factor that relates inches to centimeters. When we talked about conversion factors, we said that one inch is equal to 2.54 centimeters. I put inches here on the bottom so that they can cancel out with these inches up here on top, so you always want them on opposite levels. Doing that helps me isolate my end amount unit, which is centimeters. So then I would just multiply 32 times 2. Initially, I'll get 81 to 8 centimeters. But remember, when you're multiplying numbers together, your answer is the least number of sig figs. 32 has 26 figs in it because remember, when you don't have a decimal point, you move from right to left. You start counting once you get your first non zero number and count all the way through. So this has to sig figs. When you have a decimal point, you go the opposite way. Our first non zero number is this to you Start counting. Here you go all the way through. So we have three sig fix. So you go with the least number of sick figs, which would be too. So this would round 2 81 centimeters as our end amount. But let's say we had to do ah type of dementia analysis with way more steps. What do we do then? Well, let's say here were given 115 minutes. So that is our given amount. And we have to get two years. Years would be on our end amount to connect. Given to end. We have to utilize conversion factors. I need to cancel out minutes, so I'll put minutes here on the bottom. We know that minutes is connected to ours. We know that one hour has in it 60 minutes. And here this would represent our first conversion factor, which I'll abbreviate as conversion factor one. Sometimes it requires more than one conversion factor to get to our end amount. This is that example. Minutes would cancel out. Now I have hours. We know that hours and days are connected. We know that one day has 24 hours. This would be my second conversion factor or C f too. Hours cancel out. Now I have days and then finally we know that days and years could be connected as well. We put days on the bottom so we can cancel out with the days on top, and we know that one year has approximately 365 days. This is my conversion factor. Three. So days cancel out. And now I'm left with years. So what we're gonna have here is 1 15 on top, multiplying with a bunch of ones which doesn't change anything. But on the bottom we have multiplying 60 24 3 65. Now it's incredibly important. You know how to plug this into your calculator. My suggestion, when you have multiple things on the bottom multiplying, is to just multiply them all together and get that sum total. When we multiply 60 times 24 times 3 65 their total is 525, and we still have the 1 15 on top when we divide 1 15. By that total, we're going to get 2.19 times 10 to the negative four. We're going to stay here, put that so 2.19 times since the night before. We're going to say here that the number of sick figs within our final answer is based on our given information now are only given information was the 15. These other numbers were the ones who supplied that. Those numbers were not given to us in any way, so we can't use them to determine the number of sick fix. So again, it's all based on the information that's presented. Look at those numbers presented to you and use those to determine the number of sick fixing. Your final answer 1 15 has no decimal point. So we moved from right to left. It has in it three sig figs. So our answer should have three sick fix, which it does so 2.19 times 10 to the negative four years would be our answer here. So everything we've learned up to this point, we're gonna use in some way when it comes to dementia analysis and remember, our conversion factors are incredibly important because there are a way of connecting are given amount to our end amount. The whole point is toe cancel out units and isolate the unit that you're looking for at the end. Now that we've done these example questions, let's move on in our discussion of dimensional analysis,