WEBVTT
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first problem, we would guess that the smaller ring
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has more wood. Um, since it is all
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closer together, uh, the bigger one is more
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spread out with more would that was drilled away.
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So on the other hand, if we were able
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to squeeze the bigger ring together to make it closer
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to the size of the smaller one, it would
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appear to be about the same. Um, so
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it does appear to be about the same. And
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then for part B, what we want to do
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is check our guests. So we're gonna use the
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cylindrical shell method. We have our circle right here
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, and then we have our This is the origin
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and we have a radius right here of big are
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and then a smaller radius of little are. Then
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we want to get the height of each cylindrical shell
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and and multiplied by two. So we get that
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X squared, plus y squared equals R squared solving
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for wine. We get that y equals the square
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root. And this is the height right here at
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the height. Mhm, um y equals the square
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root of R squared minus X squared so that our
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volume is going to equal to pipe of the integral
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from our to our of the shell radius times the
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shell height, DX So what that's gonna look like
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is the shell radius of X and the show height
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, which is gonna be two times the y value
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. So two times the square root of R squared
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minus x squared dx. When we integrate this,
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we end up getting that are volume is going to
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be four thirds pi Our big R squared minus little
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r squared to the three house power. Um Then
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what we see is that this r squared minus r
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squared we've seen before because this is the same thing
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as a church squared, Um, so we can
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replace that with h squared. Thus, in doing
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so, we would, um, looking at each
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squared we have r squared plus h squared over four
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equals big R squared so R squared minus big R
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squared minus r squared. If we bring this over
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, is gonna end up giving us I'm eight spread
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over four and then we multiply by four to get
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each squared. So looking at that, we see
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that we can rewrite this as V equals four thirds
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pi of H squared over four to the three house
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. And this simplifies to be 16 Hi, h
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cute.