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The Great Pyramid of King Khufu was built of limestone in Egypt over a 20-year time period from 2580 bc to 2560 BC. Its base is a square with side length 756 ft and its height when built was 481 ft. (It was the tallest man-made structure in the world for more than 3800 years.) The density of the limestone is about $ 150 lb/ft^3 $.(a) Estimate the total work done in building the pyramid.(b) If each laborer worked 10 hours a day for 20 years, for 340 days a year, and did 200 ft-lb/h of work inlifting the limestone blocks into place, about how many laborers were needed to construct the pyramid?
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Calculus 2 / BC
Chapter 6
Applications of Integration
Section 4
Work
Baylor University
University of Nottingham
Idaho State University
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in this video, we are going to go over how to estimate the work in building a pyramid, and then we're going to estimate the number of workers it took that we think, to build the pyramid. So let's consider, let's consider our pyramid. It looks something like this. It looks something like this. Yeah, from here, what we're going to do is we're going to take a horizontal slice of this pyramid in the form of a square and we are going to draw this cross component for this cross section right here, just to make sure what it is. But we're also going to in order to figure out the area of the cross section. We need to do this in order to be able to integrate over the whole pyramid in order to find how much work it took. We need to find a couple of components. First off, we need to find the area. Yeah, we need to find the area of this pyramid of this square cross section. And in order to do that, we just need to find the length of a side. We can do that by using similar triangles. Let we know that the height of this pyramid is for 81. In doing that, we can say that the height of this. We can say that this height over here in green, up to the cross section is let's call it. Why? By default, that means that the other remaining length that is in dotted green is 4 81 minus Y. Of course the units are correct to now. Yeah, we know that the the total length of this pyramid, which I'll draw in blue, or the total the total base length is 7 56. And we can call the length of this cross section two. B. L. By similar triangles. We have that. Why? Over L. We have the Y. Over L. Is equal to 4 81 Over 7 56. In solving for L. We will get that L. Is exactly 7 56/4 81 times. Why this is the length of the square of the side of the square. So let's put that here. 7 56 divided by 4 81 times Y. So because we found the length of the side of the square, we can now just find the area of the square by squaring our side length. Yeah. Now the in order to find the differential volume of this slice. Well, just multiply the area we found by Dy. I'll just write it like that. Dy now the next step is we need to find the force that's given, we know that the force is equal to volume times density, mm. So we can were given the density of 1 50. So I can just multiply my volume by 1 50. And finally when we do this, this allows us to find the work that's done on this slice. That involves finding how high it moves. Which is what we found earlier for 81 minus Y 4 81 minus y. Times 7 56. Over 4 81 squared B. Y. Yeah. And by summing up all of the slices possible. That's an integration. So you can integrate it from the bottom to the total height of the pyramid of this entire quantity. Right. I forgot to Why here? I'm sorry. Let me put that in. All right. Mhm. Mhm. And we can do this numerically. This is not too hard to integrate. All you need to do is you just need to factor out the constants and distribute. But I'm going to elect to do that on my calculator when we do that. The work we get is going to be approximately 1.65 three times 10 to the 12th foot pounds. That's our approximation for the work in building this pyramid. Yeah. Now for the second part of the question it asks us how many workers it took to construct the pyramid, given that each labor work for 10 hours a day for 20 years, for 340 days in the year and did 200 foot pounds per hour of work into lifting the blocks. That just takes a little bit of this just takes a little bit of dimensional analysis. So let's just write the dimensional analysis here. So how many workers? Let's write that down? So you'll get that. It took 1.653 times 10. The 12th foot pounds. Let's write it like this per pyramid. I'll just write it like that. P. Y. R. For one for one person in an hour. It took them 200 ft pounds. And they did that in for 10 hours in a day. And they did 340 days in the year. Yeah. Come on. And they did that for 20 years. And as you can see all of the units will very nicely cancel out. Foot pounds will cancel. Days will cancel, years will cancel hours for hours will cancel and pyramids will cancel. Yeah. And when you do the math, this gives us approximately 1 21,000 people worked on this pyramid. And that's how you do this question.
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