WEBVTT
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So in this problem we are given a bowl that
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is 30 cm in diameter And we drop a 10
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cm diameter ball in this bowl. Heavy ball we
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begin to fill with water and the water level in
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here. Right? Is it some height? H
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. Okay. And the question is for us to
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figure out the formula on the volume of water in
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the bowl based on that depth. H. Okay
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. So first of all we know that for the
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bowl the diameter is 30 centimeters so therefore the radius
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is 15 centimeters. Okay let's assume that we do
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this right? We have the origin be at the
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very bottom of the bowl here and the X.
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And Y axis go through the middle of it.
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Just like that drawing right there. Okay then using
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the center radius form of a circle, this equation
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of this bulk is the hemisphere. It's a half
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a circle, right? Is X squared plus why
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minus 15 squared? I guess I should have put
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my X axis appear at the top 15 squared Is
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equal to 15 squared. Okay. Mhm. So
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as I've said we'll put our X axis up here
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. Okay now let's calculate this for values of X
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then. Right, so this is X squared plus
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Y squared. Actually we'll just say that X squared
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equals 225. That's 15 squared minus y minus 15
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squared. And so then X is 225-.
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Why-15 squared square to that? And this is
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going to be for zero. It will be valid
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for zero is less than or equal to Y is
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less than or equal to 15. Okay now which
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matches we can't have a height Greater than 15 cm
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because our bowls only 15cm. Right? So then
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the volume it's going to be found by rotating these
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values rotating this with respect to the Y axis in
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it. Alright just rotate those those why values around
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those X. Failures around the y axis. Okay
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so that means I end up with pie I was
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integral from 0 to H. of the square root
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of 2 25 Why minus 15 squared. And you
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square this I. R. Squared do you?
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Y. Right okay so that means then that I
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get high Times interval from 0 to H. Uh
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huh. 2 25 minus Y squared minus C.
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two b plus 30 y minus 2 25. All
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of this dy Well to 25-2. 25 cancels
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out doesn't it? Okay so that means I have
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pie times. We'll see the integral of 30 Y
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is 30 y over two minutes, y squared minus
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Why cubed over three. We're going to evaluate this
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from 0 to H. Okay well when I put
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in zero and therefore why every time that's all zeros
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that's all gonna can't drop out. And so that
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means I'm left with 15 H squared. Let me
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do it this way. Let me do it this
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way and make it clear Pie times 15 H squared
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minus H. Cube over three. Okay. And
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of course this is from zero is less cynical age
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is less than or equal to 15. So this
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is the volume in the bowl. All right.
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So similarly we do the same thing for the ball
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, right? For the ball we have diameter is
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10 cm and so The radius is five cm.
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Okay, again, doing the same thing like we
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did earlier we have the ball here for X and
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Y this way. Okay, so we write the
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equation for that circle. Well that's X squared plus
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. Why? Minus five squared is five squared?
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And so that means X is 25 minus Y-5
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Squared. Sorry, that's X squared. So the
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X. Is the square root of 25-6-5
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squared. All right, So rotate this, rotate
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with respect two the y axis on this to do
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the volume and then we have the volume of the
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ball. Is pie Times The integral from 0 to
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H. of the square root of 25 minus y
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minus five squared squared power squared. Right do you
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want? Okay so this is Pie, I'm senator
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from 0 to age of 25 minus y minus five
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squared. Do you Why? And so this is
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pie I was integral from 0 to H. Of
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25 minus Y squared plus 10 Y minus 25.
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Do you why? and 25-25 zero. So
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those cancel out. And so I end up with
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pi times the integral of 10 Y. So that's
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10 Y squared over two minus Y cubed over three
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, Evaluated from 0 to H. Okay So that
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means I have pie. We put zero in there
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for why? That's gone. So this is pi
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times five Y or not? Why it's H.
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Now? Right. Eight squared- H. Cubed
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Over three. Okay. And this is only valid
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from zero two H To 10. Right? With
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h between zero and 10 because the name of the
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ball is 10. All right. So volume the
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water in the bowl is the volume of the ball
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minus the environment. The ball takes up at any
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point at H. Right? So for zero is
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less than equal to age is less than equal 10
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. So for height of 10 cm or less than
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we have. The volume of the water is our
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two formulas together. So that's pi times 15 H
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squared- Age cubed over three minus pi times five
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H squared minus H cube over three. All right
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, well, so distribute this out in here.
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And what do we see? We'll see that this
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and this cancel out. And so we'll have yeah
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15 pi eight square-5 x eight squared. That's
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when we left with 10 pi H squared is the
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volume of the water for zero is less than equal
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to H is less than equal to 10 For the
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1st 10 cm of height. So then for 10
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is less than H is less than or equal to
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15. Right then the volume of the water is
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the volume of the ball minus the total volume of
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the ball. No, so that means I have
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pi times 15 H squared minus Age cubed over three
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minus The total volume of the ball which is 4/3
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pi r cubed by definition of a sphere. Okay
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, so since the radius is five centimeters this is
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high times 15 H squared minus H. Cute.
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Over three minus four thirds. Hi times five cubed
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. Well five cubed is 125. Okay, so
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that means I'm left with 15 pi H squared minus
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pi H cubed over three minus four times 125.
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So that's 500 Over three time spy. And so
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there's the formula right? For 10 is less than
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H is less than April 15 for when the height
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is over 10 centimetres. And this is the formula
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When were from 0 to 10 cm. Hi