The Great Pyramid of King Khufu was built of limestone in...

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 $\mathrm{ft}$ and its height when built was 481 $\mathrm{ft.}$ (It was the tallest manmade structure in the world for more than 3800 years.) The density of the limestone is about 150 $\mathrm{lb} / \mathrm{nt}^{3}$ . \begin{equation} \begin{array}{l}{\text { (a) Estimate the total work done in building the pyramid. }} \\ {\text { (b) If each laborer worked 10 hours a day for } 20 \text { years, }} \\ {\text { for } 340 \text { days a year, and did } 200 \text { ft-lb/h of work in }} \\ {\text { lifting the limestone blocks into place, about how }} \\ {\text { many laborers were needed to construct the pyramid? }}\end{array} \end{equation}

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 $\mathrm{ft}$ and
its height when built was 481 $\mathrm{ft.}$ (It was the tallest manmade structure in the world for more than 3800 years.)
The density of the limestone is about 150 $\mathrm{lb} / \mathrm{nt}^{3}$ .
\begin{equation}
\begin{array}{l}{\text { (a) Estimate the total work done in building the pyramid. }} \\ {\text { (b) If each laborer worked 10 hours a day for } 20 \text { years, }} \\ {\text { for } 340 \text { days a year, and did } 200 \text { ft-lb/h of work in }} \\ {\text { lifting the limestone blocks into place, about how }} \\ {\text { many laborers were needed to construct the pyramid? }}\end{array}
\end{equation}

Key Concepts

Energy Distribution and Labor Estimation

After computing the total work required to build a structure, matching this energy demand with the work capacity of individual laborers involves unit conversions and estimations. This concept connects the total energy needed with the available energy per worker over a specific time period, thus estimating the number of laborers required for the construction.

Density and Mass Relationship

Density links the mass of a material to its volume and is crucial when estimating the weight of construction materials. By multiplying the calculated volume of the pyramid by the density of the limestone, one can determine the total mass, which is essential for further work and energy calculations.

Integration in Estimating Work

When work varies continuously throughout a structure like a pyramid, integration is used to sum the contributions of infinitesimal elements. This approach models the lifting of material from ground level to various heights, leading to an accurate overall calculation of work done by integrating over the height or volume.

Work Against Gravity

This concept involves lifting an object against the force of gravity, which requires doing work equal to the product of the object's weight and the vertical distance it is raised. In problems of lifting large structures, calculating the work done provides an estimate of the energy or effort necessary to move materials from one elevation to another.

Volume and Geometry of a Pyramid

Understanding the geometry of a pyramid is key for determining its volume and, by extension, its mass when combined with the density of the material. The pyramid's volume can be calculated using its base area and height, allowing for the estimation of the total amount of material involved in its construction.

Transcript

00:01 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.

00:15 So let's consider our pyramid.

00:20 It looks something like this.

00:24 It looks something like this.

00:28 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 or this cross section right here just to make sure what it is.

00:46 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.

00:59 We need to find a couple of components.

01:01 First off, we need to find the area.

01:08 We need to find the area of this pyramid, of this square cross section.

01:13 And in order to do that, we just need to find the length of a side.

01:18 We can do that by using similar triangles.

01:22 Let we know that the height of this pyramid is 481.

01:30 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 y.

01:43 By default, that means that the other, the remaining length that is dotted green is 481 minus y.

01:53 Of course the units are correct too.

01:58 We know that the total length of this pyramid, which i'll draw in blue, or the total base length, is 756.

02:10 And we can call the length of this cross -section to be l.

02:16 By similar triangles, we have that y over l, we have that y over l is equal to 481 over 756.

02:39 In solving for l, we will get that l is exactly 756 over 481 times y.

02:53 This is the length of a square of the side of the square.

02:58 So let's put that here.

03:01 756 divided by 481 times y.

03:05 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.

03:23 Now, in order to find the differential volume of this slice, we'll just multiply the area we found by d .y.

03:44 I'll just write it like that, d .y.

03:47 Now, the next step is we need to find the force that's given.

03:54 We know that the force is equal to volume times density...

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