Scientific notation word problems
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A human body has 5 liters of blood, 40% of which is red blood cells. Each red blood cell has a volume of approximately 90 times 10 to the negative 15 liters. How many red blood cells are there in a human body? Write your answer in scientific notation, and round to two decimal places. So they tell us the total volume of blood in the human body. We have 5 liters. And then they tell us that 40% of that is red blood cells. So if we take the 5 liters and we multiply by 40%, this expression right here gives us the total volume of the red blood cells, 40% of our total volume of blood. Now this is the total volume of red blood cells. And we divide by the volume of each red blood cell. Then we're going to get the number of red blood cells. So let's do that. Let's divide by the volume of each red blood cell. So the volume of each red blood cell is 90 times 10 to the negative 15 liters. So let's see if we can simplify. So one thing that we can feel good about is that the units actually do cancel out. We have liters in the numerator, liters in the denominator, so we're going to get just a pure number, which is what we want. We just want how many red blood cells are actually in the body. So let's just focus the numbers here. So 5 times 40%-- well 40% is the same thing as 0.4. So let me write that down. This is the same thing as 0.4. 5 times 0.4 is 2. So our numerator simplifies to 2. And in the denominator, we have 90 times 10 to the negative 15, which definitely is not in scientific notation. It looks like it, but remember, in order to be in scientific notation, this number has to be greater than or equal to 1 and less than 10. It's clearly not less than 10. But we can convert this to scientific notation very easily. 90 is the same thing as 9 times 10, or you could even say 9 times 10 to the first. And then you multiply that times 10 to the negative 15. And then this simplifies to 9 times-- let's add these two exponents-- 10 to the negative 14. And now we can actually divide. And let's simplify this division a little bit. This is going to be the same thing as 2/9 times 1 to t over 10 to the negative 14. Well what's 1 over 10 to the negative 14? Well that's just 10 to the 14. So this right over here is just the same thing as 10 to the 14. Now you might say, OK, we just have to figure out what 2/9 is and we're done. We've written this in scientific notation. But you might have already realized, look, 2/9 is not greater than or equal to 1. How can we make this greater than or equal to 1? Well we could multiply it by 10. If we multiply this by 10, then we've got to divide this by 10 to not change the value of this expression. But let's do that. So I'm going to multiply this by 10, and I'm going to divide this by 10. So I haven't changed. I've multiplied and divided by 10. So this is equal to 20/9 times 10 to the 14th divided by 10 is 10 to the 13th power. So what's 20/9? This is going to give us a number that is greater than or equal to 1 and less than 10. So let's figure it out. And I think they said round our answer to two decimal places. So let's do that. So 20 divided by 9-- 9 doesn't go into 2. It does go into 20 two times. 2 times 9 is 18. Subtract. Get a remainder of 2. I think you see where this show is going to go. 9 goes into 20 two times. 2 times 9 is 18. We're just going to keep getting 2's. So we get another 2, bring down a 0. Nine goes into 20 two times. So this thing right over here is really 2.2 repeating. But they said round to two decimal places, so this is going to be equal to 2.22 times 10 to the 13th power.