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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 in

lifting the limestone blocks into place, about how many laborers were needed to construct the pyramid?

(a)$$

W_{\text { total }} \approx 5.31 \times 10^{13}[\mathrm{ft}-\mathrm{lb}]

$$

(b) 121545 workers were used

Applications of Integration

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Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Boston College

on this problem. We're looking at one of the great pyramids were told that it has a massive seven times 10 to the nine kilograms ah height of 146 meters and its center of mass at 1/4 of this height. We're also told that it takes 20,000 workers to build this pyramid over the course of 20 years, working 330 days per year at 12 hours per day. Lastly, we're told that these workers metabolize food energy at a rate of 300 kilocalories per hour. The first part of one says to calculate the potential energy stored on this pyramid. Still So to do that, we wanna recall that the potential energy is equal to M G multiplied by the height above our reference of the center of mass. The reference here is the ground. So we're told that this is 1/4 of its total height 1/4 and G H total mass is seven times 10 to the nine kilograms. The height is 146 meters. So this up the three significant figures gives us 2.5 times 10 to 12. Jules, the second part wants us to calculate the efficiency of lifting the stones, assuming that 1000 of the 20,000 workers are doing the lifting. So to do this, we won't recall that the efficiency is given by the output work divided by the energy used to achieve that out. But work here that's gonna be what's metabolized by the food been multiplied by 100. To get into a percentage, calculate the output work. We want to use the non conservative Virgin of Conservation of energy. This tells us that yeah, initial energy of our system plus the output work is equal to the final energy. The initial configuration of our system is no peer middle at all. So are all of the stones on the ground. So that zero no kinetic, no potential and our final energy is gonna be completely potential. So it's that 1/4 MGH showing us that our output work is equal to the stored energy in the final configuration have about the metabolized energy. But we're told that the workers metabolize food energy at a rate of 300 kilocalories per hour. So we need to find the total amount of hours that thes 1000 workers do in lifting this entire system. So let's see. We've got 1000 workers working for 20 years, working 330 days in those years and working for 12 hours in each of those days. So that's the total amount of time. In hours we have 300 kilocalories for hours. So we just need to multiply this by our total time and make sure all the units work out. So her efficiency 1/4 in G H over this rate of metabolizing food energy and we'll just call that are what's played by our total time 100. Let's see if this gets is sort. Total mass is seven. It's tender, the nine kilograms 146 meters tall. And we've got jewels up here or right out units just so we make sure we get those all squared away. Our rate is 300 kilocalories per hour. We got a total of 1000 times 28 times 330 times 12 hours. So we see that in the numerator we got jewels, denominator. We got kilocalories. So let's get our equivalents between these two everyone Kelly Calorie, We've got 4184 jewels and this hope most but by 100 for that, uh, percentage. And this, the three significant figures gets us 2.5 You 0.5 to percent. Problem also wants to comment us to comment on this low number. The slow number is telling us that a lot of the work of these 20,000 workers is going into or sorry of the workers that are doing this lifting is going into friction and in just lifting their bodies up and down. Uh, this pyramid, for the last part it wants, is to calculate the total massive food needed for these workers if we assume the diets to be 5% protein, 60% carbs and 35% fat with the average worker and needing 3600 kilocalories per day. So to do this first, you can see that our total energy needed for one day should they already that explicitly total energy for one day is equal to 3600 times 20,000 workers and that's in kilocalories. Now, in order, Thio, pull out the mass. We're gonna need to refer in to our textbook, which tells us from one of the tables that for protein. One gram of that has an energy content of 4.1 kilocalories. Carbs happen have the same energy content one gram being 4.1 kill the calories and fat being more than double of that as the energy content of 9.3. Cut the calories for one gram. So our total mass now that we know are percentages. We have 5% protein person protein, 60 percent carbs, 35% fat. We just need to take into consideration these percentages and calculating our mass ratios. So we have our total mass being called this big guy or tea. So we've got our total energy needed to sustain the day. And then we've got 5% of that being our protein which has an energy content of one gram. Her 4.1 killer crap calories. We're gonna want to put this in the kilograms. So for every one Graham, we've got 4.1 of the calories got 60% of this being carbs, which has the same energy content. One gram for every 4.1. Cut the calories and we've got 35% coming from fat, which is one gram for every 9.3 other calories here. If we cleared what r T is total energy needed for the day? And that's 3600 times 20,000. Copy this and that. You ate that whole thing again. Yeah, this whole big distraction. And this up to three significant figures gives us 14,100 kilograms. So for 20,000 workers every day, we're going to need a 14,100 kilograms of food to sustain this diet of 3600 killer calories.

University of Colorado at Boulder

Applications of Integration