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Problem 48 Hard Difficulty

The base of a circular fence with radius 10 m is given by $ x = 10 \cos t $, $ y = 10 \sin t $. The height of the fence at position $ (x, y) $ is given by the function $ h(x, y) = 4 + 0.01(x^2 - y^2) $, so the height varies from $ 3 m $ to $ 5 m $. Suppose that $ 1 L $ of paint covers $ 100 m^2 $. Sketch the fence and determine how much paint you will need if you paint both sides of the fence.

Answer

Volume$=1.60 \pi \cdot$liters

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Video Transcript

So this problem is that a work and force, They give you the height and area. That's pretty pretty much the same thing. You know, there's a frugal area under a curve in this case curve. It's like a three dimensional Cruyff. I mean, the curve in this three dimensions So still in integral height. Yes. And this is given by by this this quantity so is the curve. His paramour tries this way. So excess ten Cose I Inti why's tensai Inti? And he built his one serves its a circular for feds Uh, t goes from zero to to pie and the see whole farm. Okay. And we have this and we have to figure out Yes Ah which is dear City Square, A prostitute. Why did he square take a square room and dt And that should be true abuse. We just have science school, cause I square add up to what? And nothing having ten tt and he goes from zero to two pi for which we have tio to write out X squared minus y square replaced by this. So we have zero point zero one times ah hundred co sign squared T minus a hundred science qwerty I remember the asses ten vt So let's try to simplify this. There was a to pi So a hundred points, you're one out. Cancel. Um, maybe we put a tenant from this might be easier. Zero to two pi four. Plus, we just have co sign squared minus sign square, which is called science Tootie on TT the two pi forty plus scientist He over too. I'LL recite ou t the value of two pines here are the same So we don't really have to worry about this one. We just have the word about forty. So if we plug in to pie, we end up getting eight pi and, uh, times ten. So is eighty pie.