00:01
In this question we are asked to model the earth as three regions with a unique density, and we're going to use this model to predict the average density of the entire earth, and then we're going to compare it to the average the actual average density.
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Given the known radius and mass of the earth.
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So i've already put up on the screen.
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The actual radii in kilometers of the inner core in green outer core in blue and mantle in purple.
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One of the things we're going to need to do is express our densities as kilograms per cubic meter.
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I'm going to start with density three because you notice that we have a range of possible densities for the inner core and outer core.
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And when we get to the actual calculation part of this.
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I will tell you which value i'm going to use for my estimate.
01:06
I'm going to use one that's like in the middle of each range.
01:08
So, 4 .5 grams per cubic centimeter.
01:15
Well we need to multiply by one kilogram for every 1000 grams.
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And then, for me the easiest way to think about how many actually write a little bit differently, how many cubic centimeters are in a cubic meter is, i'm going to take my hundred centimeters and cubits, because there are 100 centimeters for every cubic meter.
01:49
So we basically end up multiplying our densities by 1000.
02:03
So you can multiply grams per cubic centimeter by 1000 to get our kilograms per cubic meter.
02:18
So, like i said, we'll do that for region two and region three or region one region two later.
02:26
So, where do we start with this.
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So we're going to, we need to remember that density equals mass divided by volume right.
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And so, in order to apply this.
02:47
We will need to.
02:49
So, the first one is easy.
02:54
We will need to subtract.
02:59
Oopsie come on.
03:02
We will need to subtract the inner core from the outer core volume.
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When we go to do its calculation.
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Oh, come on.
03:15
I just really can't seem to write neatly.
03:23
And then we will also need to subtract the inner core of, sorry, outer core volume from the mantle.
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When we do our calculation of volume like involving volumes, because, you know, we're going to find some mass, we're going to basically use the density to find the mass of each section.
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So we can get the total mass.
03:56
And then we can use the outermost radius so we need to do density equals mass and volume, that's what i messed up by volume.
04:07
We need the total mass, and then use our three to get the volume of the whole planet.
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And that will be our way that we can get the average density when we have these regions of different densities.
04:35
Okay, let's dig in with region one.
04:40
So, because we're dealing with a sphere, its volume will be four thirds pi times its radius cubed.
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And, of course, the density in region one will be equal to the mass of region one divided by the volume of region one.
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So we can say that the mass in region one will be equal to the density in region one multiplied by its volume.
05:05
So we'll take the density of region one and multiply it by four thirds pi r one cubed.
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I'm going to set up a lot of this stuff symbolically first.
05:21
Before i go worrying about numbers i should warn you about that before we go too much further.
05:27
Oops, okay.
05:28
And so now for region two.
05:32
Again, the volume in region two, that actually has this density of rock will be four thirds pi r two cubed minus volume one.
05:47
So, we can actually i'll do that one row down.
05:51
Actually, so we'll have four thirds pi r two cubed minus four thirds pi r one cube.
06:00
I'm going to factor out in one step, the four thirds pi.
06:06
And again, we're going to know that oh yeah our density of region two will be the mass of region two divided by the volume of region two.
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So we can say that our mass of region two will be equal to its density times its volume.
06:19
So, hooray, we'll take the mass of region two, who multiplied by our four thirds pi times the radius of the outer core.
06:30
Oops, cube, let me fix that sorry that was not supposed to be squared.
06:33
I lost track of my cursor, minus the radius of the inner core cubed.
06:45
I'm going to have to go down, slide down scroll down a little bit here...