WEBVTT
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So what we have here is a side view of
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a cylinder inscribed in a sphere of radius R So
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we'll have our sphere here, which looks like a
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circle. And then we have This is a cylinder
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because from a side view of cylinder just looks like
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a rectangle. So this is point A we'll call
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this point B point B. This will be point
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D and this will be point c. We know
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that point is right here in the center. We
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have our radius r and then we have this length
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right here, which will be big are this is
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a church over to? We'll call this point f
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and this point e and H is this whole long
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side right here. So we want Thio take thes
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and find the volume. We already know that the
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volume of the cylinder of a normal cylinder is pi
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r squared h Then we need another equation to relate
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this with the other variable. And that's going to
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be on the Pythagorean theorem we see that are squared
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plus h over. Um let's see we have If
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we solve for r squared, we'll get that h
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squared r squared is equal to R squared minus h
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squared over four. And that's the A squared plus
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B squared equals C squared. We have a squared
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plus b squared equals C squared. And then we
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just subtracted one of the components of the other side
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. So now we can sub substitute for r squared
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into our volume equation, so we'll put this right
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here. This is gonna go right in there.
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So now what we have is that the volume is
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equal to pi R squared minus h squared over four
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h. So now what we have is that this
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is equal to pi times R squared H minus h
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cubed over four. So now we're gonna want to
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differentiate this so v prime. Well, give us
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, um, pie times are squared minus 3/4 each
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square. When we solve for it at zero,
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we end up getting that H is equal to the
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square root of four thirds R squared. So it's
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just gonna be equal to two over rad three are
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. Then we can plug this value into R V
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double prime. We know that V double prime is
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going to equal pie times negative three over to each
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So when we plug in the value into original the
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we'll get the V equals pi. Times R squared
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two over route three are minus chew over. Route
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three are cute over four, and this is going
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back to our original equation right here. Then we
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simplify this further and we get that V is equal
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to 4/3. Route three. Hi r cubed.
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So that is going to be the volume that we
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find, um, for this particular cylinder.