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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

Yeah mm. Okay. This question asks you to derive the formula for the height of the frost. Um and that's spelled with only one R. F R. U. S. T. Um of a square pyramid. And they give you the base of the larger one is letter B. And the base of the smaller one is letter A. And the height is little H. This is the height of the frost. Um That's given so I need the volume in terms of that A. And B. And little H. So I can find the volume of the full pyramid all the way up to the top by saying one third the area of the base times the height of the whole thing which is big H plus little H. Amusing Big H. To be the height of the small one from a up a squared up. And as I said, the little H is just the difference between the height at being the height of little aid. So this is the volume of the big in volume of the small Is 1/3 the area of its base times it's small height h. So the volume of the thrust I'm I'm interested in is one third B squared big H plus little H minus one third A squared big H. Now I need to do a little bit of work about Big H. And little H. If I look at a side view of the cross section through the center and through the vertex perpendicular to the base and passing through the vertex. Just a side view. Yeah, of this triangle. The base here is be the base here is A. And the ratio of the height of the little triangle to the height of the whole will be the same as the ratio of their bases. So I know that the height of the little one Divided by the height of the Big one is equal the ratio of the basis of those two triangles over B. So big H. Times little B. Cross. Multiplying I get big H. Times B. Is equal to a. Times big H. Plus little H. And I need an expression for big H. So I get B. H -8 H equals a little h. So big H. Is little H over B minus A. So I replaced both of the big H. Is with that Manufacture out the 1/3 and leave it outside at B squared times. Big H minus whoops. Wait a minute. I forgot the A. I get little H. Times A divided by b minus a. I factored out in H. And made this b minus A. Because a H divided by the b minus A. To get this right here. Got a little crowded. Okay so I'm replacing big H. With little H. A. Over B minus a minus little H -13 A Square. Yeah tom's big H. Mhm. Okay so that becomes one third of let's see everybody has an H. In them all the terms was he B squared times H. Time is a over B -A -1 minus a squared times H times a over b minus A. So I'm gonna grab everybody. I've got so far oops. Try it again, grab everybody. Yeah and shrink a little bit. Yeah it's not working. We try again. Yeah. Yeah. Okay. Okay. Yeah. Yes. Right. Mhm. No it's not gonna work. So let's go down just a little and rewrite this piece H. When the one third factored out A squared H. A over B minus A. All right now we're cleaning up that Get volume of the frost on this 1/3 of H times A B squared over B minus a minus B squared minus a squared A. Over b minus a. So that's 1/3 times a b squared minus a. A squared over B minus a minus b squared one third H times a b squared minus a squared over b minus a minus b squared. We're getting closer and believe it or not That's 1/3. I'm going to factor the b squared minus a squared and to be minus a B plus A. So I can cancel the b minus a. S. Now I have a times B plus a minus B squared Which is 1/3 temps A squared plus a B should be plus B squared which means back here. I wrote a minus B when I met two plus b. Okay you see right here where I had an H plus H. When I filled it in down here I had an H plus H. I changed it to H minus H. By mistake. That should be A plus. Making this A plus this plus this plus in this plus this A plus. And it turns out the way it should volume of the frost. Um This 1/3. Times A squared plus A B plus B squared. How was long? Now? They asked you what happens when A equals B? If A equals B. Looking at the original picture, my age will go to zero. If A is B Then I just have a square. My volume is zero And if a approaches zero then I have one third H B squared. I just have the full pyramid. Thanks for watching.