SECTION A.2 Geometry Essentials A21 54. The Bermuda Triangle Isabella is doing research on the Bermuda Triangle which she defines roughly by Hamilton, Bermuda; San Juan, Puerto Rico; and Fort Lauderdale, Florida. On her atlas Isabella measures the straight-line distances from Hamilton to Fort Lauderdale, Fort Lauderdale to San Juan, and San Juan to Hamilton to be approximately 57 millimeters (mm), 58 mm, and 53.5 mm, respectively. If the actual distance from Fort Lauderdale to San Juan is 1046 miles, approximate the actual distances from San Juan to Hamilton and from Hamilton to Fort Lauderdale. 52. Construction A circular swimming pool that is 20 feet in diameter is enclosed by a wooden deck that is 3 feet wide. What is the area of the deck? How much fence is required to enclose the deck? 53. How Tall Is the Great Pyramid? The ancient Greek philosopher Thales of Miletus is reported on one occasion to have visited Egypt and calculated the height of the Great Pyramid of Cheops by means of shadow reckoning. Thales knew that each side of the base of the pyramid was 252 paces and that his own height was 2 paces. He measured the length of the pyramid’s shadow to be 114 paces and determined the length of his shadow to be 3 paces. See the figure. Using similar triangles, determine the height of the Great Pyramid in terms of the number of paces. Source: Diggins, Julie E, String, Straightedge, and Shadow: The Story of Geometry, 2003, Whole Spirit Press, http://wholespiritpress.com. 20' 3' In Problems 55–57, use the facts that the radius of Earth is 3960 miles and mile feet 1 5280 . = 55. How Far Can You See? The conning tower of the U.S.S. Silversides, a World War II submarine now permanently stationed in Muskegon, Michigan, is approximately 20 feet above sea level. How far can you see from the conning tower? 56. How Far Can You See? A person who is 6 feet tall is standing on the beach in Fort Lauderdale, Florida, and looks out onto the Atlantic Ocean. Suddenly, a ship appears on the horizon. How far is the ship from shore? 57. How Far Can You See? The deck of a destroyer is 100 feet above sea level. How far can a person see from the deck? How far can a person see from the bridge, which is 150 feet above sea level? 58. Suppose that m and n are positive integers with m n. > If a m n , 2 2 = − b mn 2 , = and c m n , 2 2 = + show that a, b, and c are the lengths of the sides of a right triangle. (This formula can be used to find the sides of a right triangle that are integers, such as 3, 4, 5; 5, 12, 13; and so on. Such triplets of integers are called Pythagorean triples.) Explaining Concepts 59. If the radius of a circle is doubled, does the area of the circle also double? Explain. 60. If the radius of a sphere is doubled, does the volume of the sphere also double? Explain. 61. You have 1000 feet of flexible pool siding and intend to construct a swimming pool. Experiment with rectangularshaped pools with perimeters of 1000 feet. How do their areas vary? What is the shape of the rectangle with the largest area? Now compute the area enclosed by a circular pool with a perimeter (circumference) of 1000 feet. What would be your choice of shape for the pool? If rectangular, what is your preference for dimensions? Justify your choice. If your only consideration is to have a pool that encloses the most area, what shape should you use?
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