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Egypt's Old Kingdom began in about 2700 B.C.E and lasted for 550 years. During that time, at least six pyramids were built as monuments to the life and afterlife of various pharaohs. These pyramids remain among the largest and most impressive structures constructed by any civilization. The building of the pyramids required a mastery of art, architecture, engineering, and social organization at a level unknown before that time. The collective effort required to complete the pyramids transformed Egypt into the first nation-state in the world. Of the six pyramids, the best known are those on the Giza plateau outside Cairo, and the largest of those is the Great Pyramid built by Pharaoh Khufu (or Cheops to the Greeks) in about 2550 B.C.E. With a square base of 756 feet on a side and a height of feet, the pyramid is laced with tunnels, shafts, corridors, and chambers, all leading to and from the deeply concealed pharaoh's burial chamber (see below figure). The stones used to build the pyramids were transported, often hundreds of miles, with sand sledges and river barges, by a labor force of , as estimated by the Greek historian Herodotus. Historical records suggest that the Great Pyramid was completed in approximately years.
a. To appreciate the size of the Great Pyramid of Khufu, compare its height to the length of a football field (which is yards).
b. The volume of a pyramid is given by the formula area of base height. Use this formula to estimate the volume of the Great Pyramid. State your answer in both cubic feet and cubic yards.
c. The average size of a limestone block in the Great Pyramid is cubic yards. How many blocks were used to construct this pyramid?
d. A modern research team, led by Mark Lehner of the University of Chicago, estimated that the use of winding ramps to lift the stones, desert clay, and water would allow placing one stone every minutes. If the pyramid workers labored hours per day, days per year, how long would it have taken to build the Great Pyramid? How does this estimate compare with historical records? Why did Lehner's team conclude that the Great Pyramid could have been completed with only laborers, rather than the laborers estimated by Herodotus?
e. Constructed in for the Paris Exposition, the Eiffel Tower is a -foot iron lattice structure supported on four arching legs. The legs of the tower stand at the corners of a square with sides of length feet. If the Eiffel Tower were a solid pyramid, how would its volume compare to the volume of the Great Pyramid?