60 CHAPTER R Review of Basic Concepts 106. Geometric Modeling Use the figure to geometrically support the distributive property. Write a short paragraph explaining this process. x y z 107. Volume of the Great Pyramid An amazing formula from ancient mathematics was used by the Egyptians to find the volume of the frustum of a square pyramid, as shown in the figure. Its volume is given by V = 1 3 h1a2 + ab + b22, where b is the length of the base, a is the length of the top, and h is the height. (Data from Freebury, H. A., A History of Mathematics, Macmillan Company, New York.) (a) When the Great Pyramid in Egypt was partially completed to a height h of 200 ft, b was 756 ft, and a was 314 ft. Calculate its volume at this stage of construction to the nearest thousand feet. (b) Try to visualize the figure if a = b. What is the resulting shape? Find its volume. (c) Let a = b in the Egyptian formula and simplify. Are the results the same? 108. Volume of the Great Pyramid Refer to the formula and the discussion in Exercise 107. (a) Use V = 1 3 h1a2 + ab + b22 to determine a formula for the volume of a pyramid with square base of length b and height h by letting a = 0. (b) The Great Pyramid in Egypt had a square base of length 756 ft and a height of 481 ft. Find the volume of the Great Pyramid to the nearest tenth million cubic feet. Compare it with the 273-ft-tall Superdome in New Orleans, which has an approximate volume of 100 million ft3. (Data from Guinness Book of World Records.) (c) The Superdome covers an area of 13 acres. How many acres, to the nearest tenth, does the Great Pyramid cover? (Hint: 1 acre = 43,560 ft2) (Modeling) Number of Farms in the United States The graph shows the number of farms in the United States for selected years since 1950. We can use the formula Number of farms = -0.0000280x3 + 0.00410x2 - 0.199x + 5.36 to get a good approximation of the number of farms for these years by substituting x = 0 for 1950, x = 10 for 1960, and so on, and then evaluating the polynomial. For example, if x = 10, the value of the polynomial is approximately 3.8, which differs from the data in the bar graph for 1960 by only 0.1. Evaluate the polynomial for each year to the nearest tenth. Then give the difference from the value in the graph. 109. 1950 110. 1970 111. 1990 112. 2017 h b b a a Data from U.S. Department of Agriculture. 5.4 3.7 2.8 2.4 2.1 2.2 2.2 2.0 ’50 ’60 ’70 ’80 ’90 ’00 ’10 ’17 Number of Farms in the U.S. since 1950 Farms (in millions) Year
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