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

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03:54

WZ

Wen Z.

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Video Transcript

we know the formula for point slope form is why minus y wan is m times x minus X one, which means the why is m which is the slope comes Xmas X one plus y one correct plug. And now what the problem has given us. And we see this essentially means that if we because this is a hat is the settling of a cross section and this is half of that length, we know that doubling it would give us the value we need, which is going to be SFX essentially. So doubling this gives us a minus B times h of acts must be So what essentially means that when we double that we multiplied this whole thing by two. We just crossed off all the twos. Right. Okay, now that we have this, we know that we have toe again. We're gonna be getting the volume. So V is from zero to h r squared D backs. Now we established what us is. We just figured that out. So plug in a menace, be over H of ax, plus b. Remember, it's squared. This is us. A square just means we have the squared now remember, when we integrate, we use the power rule, which means we increased the experiment by one and we divide by the new expert. So again we're dividing by the new exponents and then for terms such as B squared. As you can see, we have added a X on because that's what we do. We integrate when we have a constant just becomes the new coefficient, so plugging in, we know we have 1/3. This could be pulled out. It's the constant times H that's common in these two terms. A squared plus a B plus B squared. Remember, you're gonna need to use the foil method from algebra, which you should have learned. No, given this we know. But as a promise, Miss Fei ai is B, which means V is 1/3 h times a squared plus eight times A, which is a squared again, plus a squared like this, which simplifies to simply h a squared because it's just one plus one plus 1/3 which is 3/3 which is just one was just a J squared. That's explanation for why, and then we know if h zero there if a zero, then we have 1/3 H B Square because they've said that a a zero, As we can see, this is the pyramid volume formula with the base that is a square, so equal sides equal angles. So what this means is that we end up with V is 1/3 age times a squared, plus a B plus B squared. This is the formula. And as we said, if A's be, then we have a rectangular box with a zero, then we have a pyramid.