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We're asked to find the height of a cylinder with a volume of

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16 pi n to the fifth and a base whose radius is 2n.

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So just as a little bit of review of how you find the

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volume of a cylinder, let me draw a cylinder here.

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So if that is the top of my cylinder and that is the

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height of-- I'm not drawing it as straight as I could.

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If that is the height of my cylinder and if this cylinder

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was transparent and you could see the base, which is just

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like the top of it, It would look something like that.

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I should draw that second one with a dotted line to show you

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that you would only see it if our cylinder was transparent.

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That's our cylinder.

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The volume of a cylinder, if the height is h-- So if our

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height of our cylinder is h, and that if you had an area of

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either the top or the bottom, either the base of the top of

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it of a, the volume of your cylinder is going to be equal

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to the height times the area of the top or the base.

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Depending on if you turn this over, this would be the base.

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Times the area of the base.

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Now, what do they tell us here?

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We want to find the height.

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They say find the height.

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So we need to solve for h.

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This is what we don't know.

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Of a cylinder with a volume of 16 pi n to the fifth.

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So they tell us the volume.

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They tell us the volume is 16 pi n to the fifth.

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So that right there is 16 pi n to the fifth.

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And then they tell us a base whose radius is 2n.

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So they're not telling us the area, they're

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telling us the radius.

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They're telling us that this radius right here is 2n right

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over there.

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Now, if you know the radius of a circle, you know the area.

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This top and the bottom, these are circles.

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That's what makes it a cylinder.

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These are circles.

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So if you know the radius, you know this area.

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Area is equal to pi r squared.

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One of the first things we learn in geometry.

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And so in this situation, it would be equal to pi.

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Our radius, they tell us is 2n.

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So pi times 2n squared.

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And we know this is the same thing, so area is equal to,

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what is this? pi times 2 squared, which

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is 4 times n squared.

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4n squared.

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Or if we just rearranged it, area is

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equal to 4 pi n squared.

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So that's the area, that right over there.

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And so now we can solve for h.

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We have 16 pi n to the fifth.

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That's our volume.

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That is equal to our height times the area.

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Times the area of the base or the top of our cylinder.

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Times 4 pi n squared.

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Now, a good place to start if we're trying to solve for h or

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actually, it'll get us right there immediately, is let's

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divide both sides of this equation by 4 pi n squared.

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So you divide the right side by 4 pi n squared.

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Divide the left side by 4 pi n squared.

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You have to do it to both sides.

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On the right-hand side, these cancel out.

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That was the whole point.

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And you are left with h is equal to this thing over here.

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We can simplify this a good bit.

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We have a pi divided by a pi, so those cancel out.

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We have a 16 divided by 4, that cancels out to a 4

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divided by a 1, or just a 4.

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And then you have n to the fifth over n squared.

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So when you have an exponent in the denominator, you can

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subtract it from the exponent in the numerator.

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So this is going to be 4 right here.

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Or 4/1.

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I'll just write it as 4.

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The pi's disappeared.

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Times n to the 5 minus 2 power, which is 3.

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n to the third power.

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And we're done.

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The height of the cylinder is 4n to the third power.

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