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Problem 52 Hard Difficulty

Find the volume of the described solid $ S $.
A pyramid with height $ h $ and base an equilateral triangle with side $ a $ (a tetrahedron).

Answer

$V=\frac{\sqrt{3}}{12} a^{2} h$

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WG

Welch'S G.

December 1, 2020

Find the volume V of the described solid S. The base of S is a circular disk with radius 2r. Parallel cross-sections perpendicular to the base are squares.

Video Transcript

in this problem it is us that find the volume. We of the described solid as a pyramid of height edge. And based and is that equal little triangles with sides eight at a dry had drawn. Right So this is a this is a 3D figure we have so Mhm. We will right here for the area of a triangle. So area of equilateral triangle is what it is nothing but rude. three x 4 A square that A is this side of the triangle. Right. So this is the basic formula we know about the equally to tangle. We need to remember this always. Okay so now uh okay by similar triangle properties by similar triangle property we can say that okay Is nothing but this is equals to rule three x 4 and one minus. Bye bye edge. Woods, correct. Mhm. Yeah. Okay so it will be Through three x 4 is square. One minus bye bye edge in square. Mhm. Right okay, so now this is the area of the triangle. Yes, but to find the area of volume or like this too difficult triangle has the area but for the potato I drawn. Mhm. So for the cadre had drawn we will just find why you So volume will be one. Well you will be close to Integration of 0 to its right, this is the height. We don't wait, you can say no this is route three by uh huh. Is quest one minus. Bye bye at the square. Do a big catch split. Thanks. So this will be deep like you. Yeah thanks. So now we will right here the integration after integration Route three x 4 square is taking less common and we will be do their thing 1- My baby. Which was when this movie H by three one minus one whole few because we'll replace H by three bun kill here. Thanks. Stop the integration we're getting this is a question Blue three by ford is square. This is you know this is Plus Age Like three. Mhm. So this will give us the war numerous. The it was to root please by a court He squared into age 93. So this will finally be be questioned Room three x 12. Aah beats. Yeah. Right so this is the basically this is basic the name the volume right? Volume of. Sorry I hope you understand the concept. Thank you for watching