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Numerade Educator

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Problem 46 Hard Difficulty

Use the Theorem of Pappus to find the volume of the given solid.
The solid obtained by rotating the triangle with vertices $ (2, 3) $, $ (2, 5) $, and $ (5, 4) $ about the x-axis.

Answer

24$\pi$ units $^{3}$

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Video Transcript

in this problem. We were given coordinates off were seized for triangle and nowhere else. You spoke with serum to find you holding off object by rotating that they're on the X axis. Alright. What we notice is the term tells us that the golden is equal to circumference off the shape formed by retaining and run. The accents is multiplied by the area off our initial given triangle. So in order to find Mr Conference off that the first need to find the, um, co ordinates off the center of mass order centrally off that triangle. Now we know that central mass Central Jennifer Triangle is located at the intersection point of this medians less. I doubt this is a median. This divide that into equal length. And since this is, uh, the bottom work exes look at, otherwise he called three level and uppers. Look at that. Y equals class level. This work actual vehicle to will be located at why echoes for level. And since we're rotating that Iran D X axis on Lee, thing we need is a distance from its central to be, um, X axis. And since we know that, why central at foreign object will be it for that will be the distance that we are interested in. And if you attend this around XX is you would get such a disk. Now we can find circumference off that by just saying two point times kept till our where capital R will be. Actually, um, this is a capital R that is equal to wipe our So this is a good two pi n times for which is a pine. The weenie in the area off are given triangle and be known as area for triangle as the height times one sided, divided by two. So the hide off the triangle is this distance, which is three this side. This too. Let's divide it by two. We find the area of the triangle to be freedom. And when we just plug everything in from the therapy, we see that Bentivoglio that shape will be eight pi wealth flight buying three that is equal to then 24 hot