Chapter 14 Answers - BISD Moodle

Chapter , Lesson 56. About 1,576 times. ( ≈ 1,576.)Kochansky’s Construction.57.58. Equilateral.59. A 30°-60° right triangle.60. DA ≈ 0.5773503.(DA = = ≈ 0.5773503.)61. AE ≈ 2.4226497.(AE = DE – DA ≈ 3 – 0.5773503 = 2.4226497.)62. BE ≈ 3.1415333. (BE 2 = AB 2 + AE 2 soBE = ≈ 3.1415333.)63. It is a good approximation of π, correct tofive digits (or four decimal places).Set III (page )Tire Change.The ratio of the circumference of the larger tire tothe circumference of the smaller tire isπ ≈ 1.14; so, for each revolution of the tires,πthe truck goes 1.14 times as far as before.70 miles × 1.14 ≈ 79.8 miles. The truck would begoing about 80 miles per hour.Chapter , Lesson Set I (pages –)The area calculated in exercise for a typicalhurricane is more meaningful when comparedwith the areas of states such as Florida (,mi ) and Louisiana (, mi ), both of whichwere hit hard by Hurricane Andrew in .Another unit of circular area measure is the“circular mil,” defined as the area of a circlehaving a diameter of one “mil” (. in).Circular units of area are obviously of little valuein working with anything other than circles.Circular mils are used chiefly in the measurementof wire.Exercises through provide an example ofhow a good foundation in geometry is importantin calculus. Related rates and maximum/minimumproblems are particularly obvious examples inthat nearly every problem requires the use ofplane or solid geometry.Exercise is actually the first theorem ofArchimedes’ Measurement of a Circle: “The areaof a circle is equal to that of a right triangle inwhich one leg is equal to the radius and the otherto the circumference of the circle.”Although the result of exercises through, that in central pivot irrigation the number ofsubdivisions of a square region makes no differencein the area watered, may seem at first surprising,the reason is obvious when considered from theviewpoint of a dilation. No matter how small orlarge the cells might be, the ratio of the areas ofπeach circle and its corresponding square, ,remains constant.The use of the circle on a square grid toestimate π was first investigated by Gauss.According to David Hilbert and Stefan Cohn-Vossen in Geometry and the Imagination(Chelsea, ), Gauss “tried to determine thenumber f(r ) of lattice points in the interior andon the boundary of a circle of radius r, where thecenter of the circle is a lattice point and r is aninteger. Gauss found the value of this numberempirically for many values of n. . . . His interestwas prompted by the fact that an investigation ofthis function yields a method for approximatingthe value of π.” The details and an explanation ofthe procedure can be found on pages – ofHilbert’s book.Hurricanes.•1. 300 mi.•2. 940 mi. [c = π(300) ≈ 942.]•3. 71,000 mi 2 . [A = π(150) 2 ≈ 70,686.]•4. 15 mi. (700 = πr 2 , r =π≈ 15.)•5. 90 mi. [c = 2π(15) ≈ 94.]