Problem 34 - Calculus: Early Transcendentals

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 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?

Step-by-Step Solution

Answer

Approximately 200,000 laborers were needed.

Step 1: Calculate the Volume of the Pyramid

The volume of a pyramid is calculated using the formula \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \). Here the base is a square with side length 756 ft, so the base area is \( 756^2 \) square feet. The height is 481 ft. Thus, the volume is \( V = \frac{1}{3} \times 756^2 \times 481 \). Calculate this value to find the volume of the pyramid.

Step 2: Calculate the Mass of the Pyramid

Once the volume is known, calculate the mass of the pyramid using the density of limestone, which is \( 150 \text{ lb/ft}^3 \). The mass is given by \( \text{Mass} = \text{Density} \times \text{Volume} \). Multiply the volume from Step 1 by the density to find the mass.

Step 3: Calculate the Work Done

Work is calculated as \( W = \text{force} \times \text{distance} \). The force is equal to the weight of the limestone, which is the mass times the acceleration due to gravity (approximated as \( 32 \text{ ft/sec}^2 \)). For simplicity, use the average height (half of the pyramid's height, \( 240.5 \) ft) over which each block is lifted. Thus, \( W = \text{mass} \times 32 \times 240.5 \). Calculate this to find the total work done.

Step 4: Calculate Total Work Done Per Laborer

Each laborer does 200 ft-lb/h of work, works 10 hours a day, 340 days a year for 20 years. Thus, the total work done by one laborer is \( 200 \times 10 \times 340 \times 20 \). Calculate this to find the total work a single laborer can do over the 20 years.

Step 5: Estimate the Number of Laborers Needed

Finally, estimate the number of laborers needed by dividing the total work done (from Step 3) by the total work done by one laborer (from Step 4). This will give the approximate number of laborers required.

Key Concepts

Volume of a Pyramid

Understanding how to calculate the volume of a pyramid is essential to solve many practical problems. The formula for the volume of a pyramid is given by \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]In this case, the base area is a square because the Great Pyramid of Khufu has a square base, with each side measuring 756 ft. So, to find the base area, you simply square this length:

• Base Area = 756 ft \( \times \) 756 ft

The height of the pyramid is 481 ft. Therefore, you substitute these values into the formula to determine the volume of the pyramid. Calculating this provides insight into the amount of space the pyramid occupies, which is essential for understanding how much material was used in its construction.

Work Calculation

Work is a core concept in physics and engineering, representing energy used when a force moves an object over a distance. It's defined by the formula:\[ W = \text{Force} \times \text{Distance} \]For the pyramid, the main force in question is gravitational, which depends on the weight of the limestone blocks that form the pyramid.
To estimate the work done to construct the pyramid, we need to calculate the weight of these blocks. The force equals the mass of the blocks times the gravitational acceleration (approximated at \( 32 \text{ ft/sec}^2 \)). However, because the blocks were lifted to an average height (half the pyramid's height), the work formula adapts as follows:

• \( W = \text{Mass} \times 32 \times 240.5 \text{ ft} \)

This calculation shows the effort required to position the stone blocks in place, providing an intriguing look at ancient engineering feats.

Density and Mass

Density is a property of materials defined as mass per unit volume, making it a crucial concept in understanding physical characteristics of materials. For the Great Pyramid, the limestone density given is \( 150 \text{ lb/ft}^3 \). To find the mass of the pyramid:

• Mass = Density \( \times \) Volume

Utilizing the volume from the earlier section, you just multiply by the density. This relationship allows us to estimate the total weight of the pyramid. Determining mass is vital as it directly influences the work calculation by showing how much material was physically handled and positioned during construction.

Historic Structures

Historic structures like the Great Pyramid provide not only insights into ancient engineering and architecture but also challenge our understanding of historic labor and resourcefulness. Constructed over a 20-year span around 2580 BC, the Great Pyramid was, for a long time, the tallest man-made structure.
This monumental effort highlights the awesome capacity of ancient societies to organize labor and materials on a massive scale without modern machinery. Understanding how these structures were built can enlighten current engineering disciplines by offering lessons from innovative problem-solving and construction techniques utilized by our ancestors.

Labor Estimation

Estimating labor for large projects like pyramid construction involves looking at how much work each laborer can accomplish. Given that each laborer performs 200 ft-lb/h of work on average, and works a predetermined schedule:

• 10 hours per day
• 340 days per year
• For a period of 20 years

Calculating the total potential output from a single laborer allows us to divide the total work needed by the work each laborer can realistically deliver.
This kind of estimation is crucial for understanding resource allocation in large-scale projects, both in historic contexts and modern settings, as it directs how manpower is used efficiently to achieve monumental tasks.