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Problem 24 Hard Difficulty

A frustum of a pyramid is the part of the pyramid that lies between the base and a plane parallel to the base, as shown. Write a formula for the volume of the frustum of a square pyramid in terms of a, b, and h. (Hint: Consider the “missing” top of the pyramid and use similar triangles.)




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Video Transcript

So what's let them missing for a but pyramid? Have a height arena call H sub one, H one. So now the volume of the whole pyramid would be equal to 1/3 times. Um, the life of the base squared times each one and the volume of, um actually misspoke here I will say that the volume of the top of the pure rid would would have that high of one, cause it's just the missing part. But the volume of the whole pyramid would be 1/3 I'm a square. We would and be original height of the thrust. Um, with the missing height h sub one. So the volume of the frost, um, would be equal to the volume of the whole minus volume of the top of the pyramid. So the volume of the Freston would equal to 1/3 case where eight sub one plus H minus 1/3 B squared H sub one, the volume of the top, the going off the whole thing. Distribute the 1/3 a square to get 1/3 a squared H one plus 1/3 a squared H minus 1/3. Be square H one. No. Um, we're gonna re order some of these. So 1/3 a squared H one minus 1/3 B squared H one plus 1/3 a spared h. So we're putting our age ones together. Now let's factor out a 1/3 since everything has 1/3 in it, it's a squared, each one minus B squared H one plus a squared H. Now we can factor in H One out of the 1st 2 terms were locked with a squared minus. B squared was a squared h again. Um, we can say that the volume of the thrust him is equal to 1/3 time's a squared H plus h sub one times a a squared minus B squared. Now continuing on if we use similar triangles, we know that age of one the height of the missing part compared to the whole thing is gonna be B two a. Now, if we solve for ages of one and h So, um was cross multiply here eight times h one with equal be times H plus H one. So a times H one equals B h was B H one. If you distribute, let's get our, um H ones on the same side, so eight times h one minus B times H one equals B age. Factor out of H one in your luck with a minus bi equals B H. And so h one equals B H over a minus bi. Now I will substitute ages of one in to live bowling with the thrust in formula. So remember the bowling with a thrusting formula was 1/3 times a A squared H plus. Now H one is going to be B H over a minus. Bi times a A squared minus B squared. We can factor the A squared minus piece where'd as a minus bi an A plus B at the difference of squares. And then this should be a spared H plus the A minus. Bees cancel E. H. Times a Bus B, which is 1/3 1/3 of a squared H plus a B H plus B squared age when we distribute the be age. Now, if we factor out the height, H is in each term 1/3 h times a squared plus a B plus B squared