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Pre-Calc Exam Notes 23

Pre-Calc Exam Notes 23 - (c Inscribe a regular octagon...

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Applications and Solving Right Triangles Section 1.3 23 r 4 25. A manufacturer needs to place ten identical ball bearings against the inner side of a circular container such that each ball bear- ing touches two other ball bearings, as in the picture on the right. The (inner) radius of the container is 4 cm. (a) Find the common radius r of the ball bearings. (b) The manufacturer needs to place a circular ring inside the container. What is the largest possible (outer) radius of the ring such that it is not on top of the ball bearings and its base is level with the base of the container? 1 26. A circle of radius 1 is inscribed inside a polygon with eight sides of equal length, called a regular octagon . That is, each of the eight sides is tangent to the circle, as in the picture on the right. (a) Calculate the area of the octagon. (b) If you were to increase the number of sides of the polygon, would the area inside it increase or decrease? What number would the area approach, if any? Explain.
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Unformatted text preview: (c) Inscribe a regular octagon inside the same circle. That is, draw a regular octagon such that each of its eight vertices touches the circle. Calculate the area of this octagon. A B θ 27. The picture on the right shows a cube whose sides are of length a > 0. (a) Find the length of the diagonal line segment AB . (b) Find the angle θ that AB makes with the base of the cube. A B C D α β Figure 1.3.6 28. In Figure 1.3.6, suppose that α , β , and AD are known. Show that: (a) BC = AD cot α − cot β (b) AC = AD · tan β tan β − tan α (c) BD = AD · sin α sin ( β − α ) ( Hint: What is the measure of the angle ∠ ABD ? ) 29. Persons A and B are at the beach, their eyes are 5 ft and 6 ft, respectively, above sea level. How many miles farther out is Person B’s horizon than Person A’s? (Note: 1 mile = 5280 ft)...
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