678 CHAPTER 9 Polar Coordinates; Vectors 59. Actual Speed and Direction of a Swimmer A swimmer can maintain a constant speed of 5 miles per hour. If the swimmer heads directly across a river that has a current moving at the rate of 2 miles per hour, what is the actual speed of the swimmer? (See the figure.) If the river is 1 mile wide, how far downstream will the swimmer end up from the point directly across the river from the starting point? Current Swimmer's direction Direction of swimmer due to current 60. Static Equilibrium A weight of 2000 pounds is suspended from two cables, as shown in the figure.What are the tensions in the two cables? 408 308 2000 pounds 61. Computing Work Find the work done by a force of 5 pounds acting in the direction ° 60 to the horizontal in moving an object 20 feet from ( ) 0, 0 to ( ) 20, 0 . 62. Braking Load A moving van with a gross weight of 8000 pounds is parked on a street with a °5 grade. Find the magnitude of the force required to keep the van from rolling down the hill. What is the magnitude of the force perpendicular to the hill? The Chapter Test Prep Videos include step-by-step solutions to all chapter test exercises. These videos are available in MyLab™ Math. In Problems 1–3, plot each point given in polar coordinates. 1. π ( ) 2, 3 4 2. π ( ) − 3, 6 3. π ( ) −4, 3 4. Convert ( ) 2, 2 3 from rectangular coordinates to polar coordinates θ ( ) r, , where θ π > ≤ < r 0 and 0 2 . In Problems 5–7, convert the polar equation to a rectangular equation. Graph the equation. 5. = r 7 6. θ = tan 3 7. θ θ + = r r sin 8 sin 2 Chapter Test In Problems 8 and 9, test the polar equation for symmetry with respect to the pole, the polar axis, and the line θ π = 2 . 8. θ = r cos 5 2 9. θ θ = r 5sin cos2 In Problems 10–12, perform the given operation, where π π π π ( ) ( ) = + = + z i w i 2 cos 17 36 sin 17 36 and 3 cos 11 90 sin 11 90 . Write the answer in polar form and in exponential form. 10. ⋅ z w 11. w z 12. w5 13. Find all the complex cube roots of − + i 8 8 3. Then plot them in the complex plane. In Problems 14–18, ( ) = P 3 2, 7 2 1 and ( ) = P 8 2, 2 2 . 2 14. Find the position vector v equal to PP . 1 2 15. Find v . 16. Find the unit vector in the direction of v. 17. Find the direction angle of v. 18. Write the vector v in terms of its vertical and horizontal components. In Problems 19–22, = + =− − =− + v i j v i j v i j 4 6 , 3 6 , 8 4 , 1 2 3 and = + v i j 10 15 4 19. Find the vector + − v v v 2 . 1 2 3 20. Which two vectors are parallel? 21. Which two vectors are orthogonal? 22. Find the angle between the vectors v1 and v .2 In Problems 23–25, use the vectors = − + u i j k 2 3 and = − + + v i j k 3 2 . 23. Find ×u v. 24. Find the direction angles for u. 25. Find the area of the parallelogram that has u and v as adjacent sides. 26. A 1200-pound chandelier is to be suspended over a large ballroom; the chandelier will be hung on two cables of equal length whose ends will be attached to the ceiling, 16 feet apart. The chandelier will be free-hanging so that the ends of the cable will make equal angles with the ceiling. If the top of the chandelier is to be 16 feet from the ceiling, what is the minimum tension each cable must be able to endure?
RkJQdWJsaXNoZXIy NjM5ODQ=