WEBVTT
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This is Giancoli Answers
with Mr. Dychko.
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The density of the Earth is
the Earth's total mass
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divided by its total volume.
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Now its total mass with this model of
it consisting of three layers
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is gonna be the mass of the inner core
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plus the mass of the outer core
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and then plus the mass of the mantle.
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And mass of the inner core will be
the inner core's density times
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the inner core volume;
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mass of the other core will be its density
times its volume.
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And when we calculate
this outer core volume,
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we are gonna be taking the volume of this
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sphere of radius r subscript o
for outer core
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and subtracting away the volume of
this sphere of the inner core.
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So we'll just be taking the volume of
the thick shell
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and then same for the mantle:
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when we find the mantle volume, we'll take
the volume of the sphere
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with radius r mantle, r m,
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and then subtracting away the volume of
this sphere here
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with radius r o for outer core.
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And when we take the difference between
these volumes,
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it's four-thirds πr m cubed minus
four-thirds πr o cubed
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and this four-thirds π is a common factor
we can just factor out
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and it turns out that you can factor it out
of every single thing here
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so on this line here, I wrote
four-thirds π times all of it
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which wasn't even necessary because
on the bottom,
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it turns out the total volume of the Earth
has a four-thirds π factor in it as well
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so these four-third's π
would cancel anyway
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but I have already punched it into my
calculator with four-third's on the top
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so let's keep it on the bottom too.
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Now what else can I say?
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Well, we have some unit issues because
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the radii we are given are in
units of kilometers
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so we always have to write in
times 10 to the 3 meters
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and then cube it.
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So we have the inner core density—
13000 kilograms per cubic meter—
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times the inner core volume.
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So we are times'ing this by the volume
but if you can get the idea
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that we have this four-third's
π factored out
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and we are just multiplying by the inner
core radius here cubed.
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So that's 1220 times 10 to
the 3 meters cubed
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plus the density of the outer core times
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the difference in the cubes of the radii
between outer core and inner core
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and on the calculator, it looks like this
when you have this capital E notation
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it's always a good thing to do
scientific notation
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with this capital E at second
function comma here
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because when you go to the power of 3
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that exponent 3 is applied intelligently
to everything here
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this entire number considers this
1220E3 as a single number
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as it should—that's your intention—
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whereas if you typed in 1220
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with a multiply sign and then
10 to the power of 3
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and then did exponent 3 that exponent 3
would apply only to a part of the number.
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So for example, if you did 1220
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times 10 to the power of 3
and then go cubed,
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this would be bad;
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it would actually be 1220 times
10 to the power of 9
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or 10 to the power of 27 is what
it would turn out to because
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because 3 cubed is 27 so yeah that's bad.
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Always use this capital E notation for
scientific notation because then
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it groups the two parts:
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the times 10 to the part and the other part
together as a single number.
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Okay!
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Other than that, everything else is
just plug and chug
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punching in numbers here to get our answer
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and our answer is 550 kilograms
per cubic meter.
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So if you consider the Earth
to have only 1 layer
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and we take its total mass divided by
its total volume
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what we end up with here is 5.98 times
10 to 3 kilograms
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and then divide that by four-third's π
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times 6380 times 10 to the 3 cubed
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and you end up with the density of...
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(woah, something went wrong there)
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that 5.98 times 10 to the 24
that's the problem
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24, getting all these cubes
on the mind here
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so many cubes everywhere
I wrote a cube there instead.
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Okay so let's try that again.
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We go second function enter
to get the last entry
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and then over here, we can insert the 24;
everything else looks good.
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That's more realistic; we expected something
close to the 5500 we had before.
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So that's 5497.3 kilograms per cubic meter.
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And then the percent difference will be
what we just found—
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5497.3 kilograms per cubic meter—
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minus the 5500 from before
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and that's the absolute difference in
the numerator and then divide by
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the value we are comparing against
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and we end up with a very small number
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that's about...
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well, as a percent
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so we have to multiply by 100
to get percent here
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that's negative 0.05 percent off
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of the density calculated with a model
with three layers.