WEBVTT
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This is Giancoli Answers
with Mr. Dychko.
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So this question involves a lot of algebra
so fasten your seat belt as we
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wend our way through all that stuff there
but hopefully I'll make it make sense.
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So we have the specific gravity of the body,
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it's the body's density divided by
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density of water, that's the definition
of specific gravity;
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so we take that total mass of the body
and divide by
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the total volume to get the density of
the body and then divide that by
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density of water so this is specific
gravity here as well
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and we are told that that
equals the letter X.
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So we need some way to get this
little f involved here because
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our final answer has to be to show
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that you know, f times a 100
which will be
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percent body mass equals some stuff
that we are showing there—
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495 over X minus 450.
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So this f has to be involved here somehow
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so we are gonna make it appear
within this volume part.
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So we know volume is the volume of
the fat in the body plus
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the volume of the fat-free
part of the body
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and the volume of the body consisting
of fat is the mass that is fat divided by
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the density of fat
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plus the mass that is fat free divided by
the density that is fat free.
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Substituting you know, you have
the density definition is
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mass over volume and you can
solve for volume by
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multiplying both sides by
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V and dividing both sides by ρ
and you end up with
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V on the left equals mass
divided by density.
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So that's what I'm substituting here:
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mass divided by density in place of volume.
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And then further we can say that m f
divided by m is f
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because f represents the fraction of
the total mass which is fat
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so the mass that is fat divided by
total mass is f
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and we can rearrange this by
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multiplying both sides by total mass
and we get
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mass of the body which is fat is this
fraction that is fat
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times the total mass and we are gonna
substitute that in
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for m f here—that's what
I did in that line—
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and then mass that is fat-free has to be
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1 minus the fraction that contains fat
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times the total mass.
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I guess to explain that I would say
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m f divided by m plus m ff
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or m has to equal 1 because
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this makes up a 100 percent of all
mass in the body so
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m ff has to equal 1 minus m f over m
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all multiplied by m
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but this thing right in here is f
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and so we have 1 minus f times m
is m ff as shown here.
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Okay.
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So plug each of these things
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into these numerator's in this
volume expression
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and we have the total volume
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that consist of fat plus volume
that is fat-free is
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f times m over density of fat plus
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1 minus f times m over the density of
the fat-free material
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and then plugging that all in for
total volume in this expression
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I guess this part of it anyway
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plugging in for V there in red
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I have total mass divided by
density of water
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times this whole business for
the total volume
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equals x and now... (what do I say here?)
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We want to solve for f which is
a nasty thing because
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it's buried within this denominator
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and well, first of all, let's say
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multiply both sides by 1 over X
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and then that gets rid of the X
on the right side
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and leaves us with an X on the left side
so we have m over X
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and then multiply both sides also
by this denominator
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so that on the left side, it disappears
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and on the right side, it's the only thing
that's there because
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X over X makes 1 and then we are left
only with 1 times this denominator
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and then switch the sides around too
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so that we have this unknown stuff on the left.
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So we have m over X equals
this whole denominator
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and then I expanded the denominator too
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so I multiplied this ρ w into
each of the terms
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so we have ρ w times f times m
over ρ f
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plus 1 minus f times ρ w times m
over ρ ff equals m over X.
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That might be the hardest algebra step
in this sequence
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because I kinda did a couple things at once
there but hopefully that made sense.
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Some teachers refer to what I did here
as cross-multiplying;
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you can multiply... even that's not
a very good way to explain it
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never mind the cross-multiplying.
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So.. there. Let's go to the next line!
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Let's expand this stuff and multiply
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both these terms in this bracket by
ρ w times m
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so we have 1 times ρ w times m
over ρ ff
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and by the way, the m's canceled...
(okay there we go) so there's no m
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it's a common factor on both sides so
we can divide both sides by it
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and it's gone
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so we have ρ w times 1 over ρ ff
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so that's just ρ w over ρ ff
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minus f times ρ w over ρ ff
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and I'm writing them as
separate terms here
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and never mind keeping them as one
fraction, make them two fractions
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because we wanna collect the two terms
that contain the f.
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So we do that in the next line and
we have f times ρ w
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being a common factor for
both these terms
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and multiplied by, you know, what ends up
being a 1 here over ρ f
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minus 1 over ρ ff
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and then this term got moved
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to the right hand side by subtracting it
from both sides so it's ρ w over ρ ff.
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And then write this as a single fraction
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so common denominator makes this
ρ ff over ρ ff,
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multiply this by ρ f over ρ f
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and you have ρ ff minus ρ f over
ρ f times ρ ff.
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That's all I did in that line
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and then multiply both sides by ρ f
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times ρ ff so we have ρ f times
ρ ff over X
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and then it's just the ρ f when you
multiply this term by that
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denominator because the ρ ff's cancel
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and so you are just left with ρ f
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and then we are also dividing both sides by
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ρ w times bracket ρ ff minus ρ f
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and so that's where this comes from
that's just, you know, copied there
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but whereas over here,
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it's missing the ρ w because
it cancels with this one
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and so it's just ρ ff minus ρ f.
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And finally, we plug in some numbers
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and it all handily works out to
what we needed to.
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So we have density of fat is 0.90 grams
per centimeter cubed
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times 1.10 grams per centimeter cubed
for the fat-free material
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divided by X times 1 gram per centimeter
cubed—density of water—
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times a difference between those
fat free and the fat density
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minus 0.90 divided by 1.10 minus 0.90
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and you end up with 4.95 over X minus 4.5
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that's the fraction which is fat
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and if we want the percent, we have to
multiply both sides by 100
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and so this gives a percent
which is body fat
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which will be 495 over X minus 450.