Question

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 manmade structure in the world for more than 3800 years.) The density of the limestone is about 150 lb/ft3. (a) Estimate the total work done in building the pyramid. (Round your answer to three decimal places.) × 1012 ft-lb (b) If each laborer worked 10 hours a day for 20 years, for 340 days a year, and did 250 ft-lb/h of work in lifting the limestone blocks into place, about how many laborers were needed to construct the pyramid?

Answer 1

(a) 1.653×10¹² ft·lb

(b) 97,229 laborers

Explanation:

You want an estimate of the work done to build the Great Pyramid, given it is 756 ft square and 481 ft high and made of limestone of density 150 lb/ft³. You also want to know the number of laborers required if each does 250 ft·lb/h of work for 10 h/day, 340 day/yr for 20 years.

(a) Work

The number of pounds of limestone required to build the pyramid is the product of the pyramid volume and the density of the stone. The volume is ...

V = 1/3b²h . . . . . where b is the length of one side, and h is the height

Using the given dimensions, we find the weight of the pyramid to be about ...

M = V·ρ

M = 1/3·(756 ft)²(481 ft)·(150 lb/ft³) = 13,745,440,800 lb

The work required to move this mass to a height equal to the center of mass is ...

W = M·h

W = (1.3745·10¹⁰ lb)(1/4·481 ft) ≈ 1.653×10¹² ft·lb

The work required to put the stone in place for the Great Pyramid is about 1.653×10¹² ft·lb.

(b) Work force

The work force required to provide this amount of work is estimated to be ...

`1.653*10^(12)\text{ ft-lb}*\frac{1\text{worker-h}}{250\text{ ft-lb}}*\frac{1\text{ day}}{10\text{ h}}*\frac{1\text{ yr}}{340\text{ day}}*\frac{1}{20\text{ yr}}\\\\\approx\boxed{97,\!229\text{ workers}}`

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Additional comment

You can go to the trouble to write an integral for the work done to lift increments of mass to the different heights. The end result is equivalent to lifting the entire mass to the height of the center of mass. For a square pyramid of uniform density, the center of mass is above the base a distance of 1/4 of the total height.

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