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Problem 50 Hard Difficulty

Find the volume of the described solid $ S $.
A frustum of a pyramid with square base of side $ b $, square top of side $ a $, and height $ h $

What happens if $ a = b $? What happens if $ a = 0 $?

Answer

$V=\frac{1}{3} h\left(a^{2}+a b+b^{2}\right)$

Discussion

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BS

Brad S.

September 23, 2020

What is frustum of pyramid?

EG

Erica G.

September 23, 2020

As far as I know Brad in geometry, a frustum is the portion of a solid (normally a cone or pyramid) that lies between one or two parallel planes cutting it. It is formed by a clipped pyramid; in particular, frustum culling is a method of hidden surface

JM

Julia M.

September 23, 2020

What is meant by coefficient?

DF

Doug F.

September 23, 2020

Hey Julia I got you with this one. A coefficient is usually a constant quantity, but the differential coefficient of f is a constant function only if f is a linear function. When f is not linear, its differential coefficient is a function, call it f?, de

EV

Eric V.

September 23, 2020

What is foil method in algebra?

SH

Sarah H.

September 23, 2020

How's the weather Eric? "A technique for distributing two binomials. The letters FOIL stand for First, Outer, Inner, Last. First means multiply the terms which occur first in each binomial. Then Outer means multiply the outermost terms in the product." H

Video Transcript

Okay, We're gonna find the volume of a fresh stream of the pyramid. So that means like it chopped the top off, and it's what's left over at the bottom. So in this pyramid, it's a square pyramid. The base is be on each side, and the top is a on each side, and the height is H. So if we cut it, then we get each cut will be a square. Okay, so each slice is a square. Okay, so we're gonna put it in the X Y plane. So here's B this one right here. And then this a is up. But this is like the shadow of it. If I held the light over, so I tilted it on its side and I'm shining a light down on it. The reason I'm doing that is so because I know the height from the bottom Thio from the bottom to the top is h So this point its name right here. H comma zero. Okay, this points name right here. H comma. One half a and this one 01 half b. And then here's the little square I made. So if this points name is X y then the length here is to y. And so the area of the slice to why times to why? Or four y squared. Okay, so now we're gonna integrate with respect to X, though, because we're gonna go from zero to H. So we gotta figure out how X and Y related. So they're on this line. So the slope of the line is one half a minus one half B over H minus zero. So one half a minus bi over H or a minus bi over to H. So why is a minus bi over to h X plus B over two? Because that's the Y intercept right there. Okay, Now we're just gonna add the slices up from zero h. Okay, so the mime is zero to h area of each slice for y squared. So four times a minus bi over to h x plus B over to squared DX. Okay, I want to do a substitution here will let you be a minus bi over to h X plus b over to then do you as a minus bi over to h d. X. And if X equals H u equals MOSB over two plus B over to which is a over to. And if x is zero, um, you is be over to. All right. So I need a name minus B over to age from a Put that in a minus bi over to h and put it out to make up for work. So I get to h over a minus bi in a girl Be over to toe over to you squared d'you to h over a minus bi You cubed over three from a over to, uh, I said it backwards. Be over to day over to So that's two thirds times H over a minus. Bi times. Okay, I took this three out, so it'll be a cute Oh, I forgot something. I forgot this four right here. Four y squared. I just Oh, no. There it is. I put it in there. I just forgot it right here. Okay, They're just this four. So this makes eight here. That makes eight here. So if eight thirds h ovary minus b and then I get a cubed over eight minus b cubed over eight. So I have eight thirds h a minus B times 18 a cube minus be cute. Okay. These AIDS air gone. Okay. Now I noticed that I can factor a Cuban eyes be cube, so I get one third h over a minus. Bi times. Defectors in the A minus bi times a A squared plus a B plus B squared. They responded, speeds canceled. So you get one third h a squared plus a B plus B squared, which is the volume of the frustrate, um, of the pyramid. Hope I'm not quite done yet. What if a equals B? Okay, well, if a equals B, then what? Do you have a picture off? Well, if a and B are the same, then it should be just a rectangular box. All right, so let's just plug in. Be for a so volume would be one third h b squared, plus B squared plus B squared. So three B squared. So H b squared volume of a box. That's B B and height H. Okay. Now what if what they say next? What if a equals zero? Yeah. What if a equals zero? Well, then that would be Ah, whole pyramid. All right. So let's see what happens if a equals zero, you get volume is one third h time. Zero squared plus zero plus B squared. So one third H B squared, which is the area are the volume of Ah, square pyramid. There you go.