Problem 56 Hard Difficulty

Find the volume of the described solid $ S $.
The base of $ S $ is the triangular region with vertices $ (0, 0) $, $ (1, 0) $, and $ (0, 1) $. Cross-sections perpendicular to the y-axis are equilateral triangles.

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01:59

WZ

Wen Z.

01:37

Amrita B.

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Video Transcript

were given a solid S. And we're asked to find the volume of this solid. We're told the base of S. Is a triangular region with overdoses. Who cares? I don't know. There used to be 00 10 01 In the cross sections perpendicular to the Y axis are equal lateral triangles. It might help to actually draw this. So we have our X. And Y. Axis. Uh huh. The river is is at 00 10101. Yeah, suck my dad. And then we have cross sections perpendicular the Y axis that are equal lateral triangles. Look something like this. Imagine is coming out of the plane of the page. Now mm He's just he's rich now you know he's still following him. Hell yeah. Which? No I guess headache. We want to find the area of each of these equilateral triangles. To do this we need to find the length of a side of an equilateral triangle. And to do this, redefined what curves bound this red region. But we have the curves X equals zero of course and Y equals zero. And then we have the line between the points 01 and 10 This is the line Y equals negative X. Plus one. And therefore the length of a side of a triangle which I'll call s is simply uh we can solve this curve for X and get X equals one minus Y. Not that you can keep came out. Yeah. Yeah. And it I don't I don't remember. We are just like yes, who's the new Orleans guy with that song about? I'll be fine. It sounded in my mind that because it was a thing where it was like there's another song called Let Me Find Out By is that Yeah, the Fifth World. Therefore we should be Let me find out your old man. We're saying that was that's a sign that something Mark, find out your families who have it's been me. Right? So the side is actually Uh just one -Y. And therefore the area my notes where there was between the notes of the equilateral triangle. This is going to be one half times the base which is one of these sides, times the height. And for an equilateral triangle skyscraper, we divided up into a right triangle. Like this, we have S over two squared plus the height. H squared equals S. Squared. So that H. Squared equals a nancy. The connotation 3/4 S. Squared and therefore H. Equals Route 3/2 s. And so we have one half S. Times route 3/2 S. And so as a function of why the area A. Of Y is one half times one minus Y. Times route 3/2, of course times one minus Y again, which is root 3/4, 1 minus y squared and therefore the volume is the integral. Overall the triangles. This is from y equals zero To y equals one of the area A. Of Y. Yeah. Dy well, two. So this is route 3/4 times the integral from 0 to 1 of one minus Y squared. Dy. If you evaluate this integral, Eventually get root 3/12 and voices.