Chapter Review 595 34. Correcting a Navigational Error A sailboat leaves St. Thomas bound for an island in the British West Indies, 200 miles away. Maintaining a constant speed of 18 miles per hour, but encountering heavy crosswinds and strong currents, the crew finds, after 4 hours, that the sailboat is off course by ° 15 . (a) How far is the sailboat from the island at this time? (b) Through what angle should the sailboat turn to correct its course? (c) How much time has been added to the trip because of this? (Assume that the speed remains at 18 miles per hour.) St. Thomas British West Indies 158 200 mi 35. Approximating the Area of a Lake To approximate the area of a lake, Cindy walks around the perimeter of the lake, taking the measurements shown in the figure. Using this technique, what is the approximate area of the lake? [Hint: Use the Law of Cosines on the three triangles shown and then find the sum of their areas.] 508 1008 100' 125' 50' 50' 70' 36. Finding the Bearing of a Ship The Majesty leaves the Port at Boston for Bermuda with a bearing of ° S80 E at an average speed of 10 knots. After 1 hour, the ship turns ° 90 toward the southwest.After 2 hours at an average speed of 20 knots, what is the bearing of the ship from Boston? 37. Frictional Force A box sitting on a flat surface has a coefficient of static friction of μ = 0.3. s If one end of the surface is raised, static friction prevents the box from sliding until the force of static friction is overcome.The critical angle at which the box begins to slide, θ ,c can be found from the equation θ μ = tan . c s (a) What is the critical angle for the box? (b) If the box is 5 ft from the pivot point, at what height will the box begin to slide? See the figure. 38. Frictional Force (See Problem 37.) Once the box begins to slide and accelerate, kinetic friction acts to slow the box with a coefficient of kinetic friction μ = 0.1. k The raised end of the surface can be lowered to a point where the box continues sliding but does not accelerate. The critical angle at which this happens, θ ′, c can be found from the equation θ μ ′ = tan . c k (a) What is this critical angle for the box? (b) If the box is 5 ft from the pivot point, at what height will the box stop accelerating? 39. Simple Harmonic Motion An object attached to a coiled spring is pulled down a distance = a 3 units from its rest position and then released. Assuming that the motion is simple harmonic with period = T 4 seconds, find a model that relates the displacement d of the object from its rest position after t seconds. Also assume that the positive direction of the motion is up. In Problems 40 and 41, the displacement d (in feet) of an object from its rest position at time t (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its rest position? (c) What is the time required for one oscillation? (d) What is the frequency? 40. ( ) ( ) = d t t 6 sin 2 41. π ( ) ( ) = − d t t 2 cos 42. Damped Harmonic Motion An object of mass = m 40 grams attached to a coiled spring with damping factor = b 0.75 gram second is pulled down a distance = a 15 centimeters from its rest position and then released. Assume that the positive direction of the motion is up and the period is = T 5 seconds under simple harmonic motion. (a) Find a function that models the displacement d of the object from its rest position after t seconds. (b) Use a graphing utility to graph the equation found in part (a) for 5 oscillations. 43. Damped Motion The displacement d (in meters) of the bob of a pendulum of mass 20 kilograms from its rest position at time t (in seconds) is given as π ( ) ( ) = − − ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ − d t e t 15 cos 2 5 0.36 1600 t 0.6 40 2 (a) Describe the motion of the object. (b) What is the initial displacement of the bob? That is, what is the displacement at = t 0? (c) Graph the function d using a graphing utility. (d) What is the displacement of the bob at the start of the second oscillation? (e) What happens to the displacement of the bob as time increases without bound? 44. Graph ( ) = + y x x 2 sin cos 2 by adding y-coordinates. u 5 Pivot point
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