Anderson Video - Estimating with Large Numbers

Professor Anderson
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>> Hello class, Professor Anderson here. Let's talk about the area problem and conversion of units. And the question here in this problem is how many dollar bills does it take to cover the United States? OK, so we're going to stack individual dollar bills all over the United States including Alaska and Hawaii and cover the entire U.S., how much does that take? All right so the first thing we need to know is how do those two areas relate? Well, if we think about the area of the USA that's a pretty big number. We're going to probably need a lot of dollar bills to cover it. So, let's call that number of dollar bills big N and we have to multiply that by the area of a dollar bill, OK? So what is the area of the USA? Well, this you can just look up online and we did that a second ago and we got 9.8 million square kilometers, OK? 9.8 million square kilometers. And if you look up the area of a dollar bill, you know how big a dollar bill is. What's the area of a dollar bill? It is around approximately 100 square centimeters. So if you want to know how many it's going to take, we can just look at this equation right here and we can say N is the area of the USA, divided by the area of the dollar bill. OK, and we know those numbers, right? So we can put them in. Now we can approximate this, OK? 9.8 times 10 to the 6, that is pretty close to 10 times 10 to the 6. 10 times 10 to the 6 is 10 to the 7. 100 square centimeters down in the bottom -- that is 10 to the 2 square centimeters. Now I could just divide those numbers but the units don't work out quite right. We got to get the same units on the top and the bottom. So why don't we convert both of those to meters. If we do that, what do we get? We get 10 to the 7 times a kilometer is in fact 10 to the 3 meters and we're going to square that whole thing. Down in the bottom we have centimeters. How big is a centimeter? Well, there's a hundred centimeters in a meter, so that is 10 to the minus 2 meters, right? A centimeter is much smaller than a meter, it's a hundredth of a meter, and so we got to put that number in right there. And so now look what happened. We've got a bunch of numbers that we need to multiply. We've got 10 to the 7 and we've got 10 to the 3 which is squared so that's a 10 to the 6. And down in the bottom we have a 10 to the 2. But then we have a 10 to the minus 2 squared, which becomes a 10 to the minus 4, OK? And so if we run all these numbers what do we get? Let's make a little room over here -- We get N equals the top -- 10 to the 7 and 10 to the 6 -- that is 10 to the 13. And in the bottom we had 10 to the 2 and 10 to the minus 4, so that's a 10 to the minus 2. And if I have 10 to the 13 over 10 to the minus 2, that becomes 10 to the 15. How many dollars do you need? You need 10 to the 15 dollars, which is a pretty big number. All right? OK, hopefully that's clear, if not comes see me in my office. Cheers.
>> Hello class, Professor Anderson here. Let's talk about the area problem and conversion of units. And the question here in this problem is how many dollar bills does it take to cover the United States? OK, so we're going to stack individual dollar bills all over the United States including Alaska and Hawaii and cover the entire U.S., how much does that take? All right so the first thing we need to know is how do those two areas relate? Well, if we think about the area of the USA that's a pretty big number. We're going to probably need a lot of dollar bills to cover it. So, let's call that number of dollar bills big N and we have to multiply that by the area of a dollar bill, OK? So what is the area of the USA? Well, this you can just look up online and we did that a second ago and we got 9.8 million square kilometers, OK? 9.8 million square kilometers. And if you look up the area of a dollar bill, you know how big a dollar bill is. What's the area of a dollar bill? It is around approximately 100 square centimeters. So if you want to know how many it's going to take, we can just look at this equation right here and we can say N is the area of the USA, divided by the area of the dollar bill. OK, and we know those numbers, right? So we can put them in. Now we can approximate this, OK? 9.8 times 10 to the 6, that is pretty close to 10 times 10 to the 6. 10 times 10 to the 6 is 10 to the 7. 100 square centimeters down in the bottom -- that is 10 to the 2 square centimeters. Now I could just divide those numbers but the units don't work out quite right. We got to get the same units on the top and the bottom. So why don't we convert both of those to meters. If we do that, what do we get? We get 10 to the 7 times a kilometer is in fact 10 to the 3 meters and we're going to square that whole thing. Down in the bottom we have centimeters. How big is a centimeter? Well, there's a hundred centimeters in a meter, so that is 10 to the minus 2 meters, right? A centimeter is much smaller than a meter, it's a hundredth of a meter, and so we got to put that number in right there. And so now look what happened. We've got a bunch of numbers that we need to multiply. We've got 10 to the 7 and we've got 10 to the 3 which is squared so that's a 10 to the 6. And down in the bottom we have a 10 to the 2. But then we have a 10 to the minus 2 squared, which becomes a 10 to the minus 4, OK? And so if we run all these numbers what do we get? Let's make a little room over here -- We get N equals the top -- 10 to the 7 and 10 to the 6 -- that is 10 to the 13. And in the bottom we had 10 to the 2 and 10 to the minus 4, so that's a 10 to the minus 2. And if I have 10 to the 13 over 10 to the minus 2, that becomes 10 to the 15. How many dollars do you need? You need 10 to the 15 dollars, which is a pretty big number. All right? OK, hopefully that's clear, if not comes see me in my office. Cheers.