650 CHAPTER 9 Polar Coordinates; Vectors 98. Removing a Stump A farmer wishes to remove a stump from a field by pulling it out with his tractor. Having removed many stumps before, he estimates that he will need 6 tons (12,000 pounds) of force to remove the stump. However, his tractor is only capable of pulling with a force of 7000 pounds, so he asks his neighbor to help. His neighbor’s tractor can pull with a force of 5500 pounds. They attach the two tractors to the stump with a ° 40 angle between the forces, as shown in the figure. 40° 5500 lb 7000 lb (a) Assuming the farmer’s estimate of a needed 6-ton force is correct, will the farmer be successful in removing the stump? (b) Had the farmer arranged the tractors with a ° 25 angle between the forces, would he have been successful in removing the stump? 99. Computer Graphics The field of computer graphics utilizes vectors to compute translations of points. For example, if the point ( ) −3, 2 , represented by vector = 〈− 〉 u 3 2 , , is to be translated by = 〈 〉 v 5, 2, then the new location will be ′= + =〈− 〉+〈 〉=〈 〉 u u v 3, 2 5, 2 2, 4 . So, the point ( ) −3, 2 is translated by v to ( ) 2, 4 as shown in the figure. (a) Determine the new coordinates of ( ) − 3, 1 if it is translated by = 〈− 〉 v 4, 5. (b) Illustrate this translation graphically. x 25 5 y 5 25 (2, 4) (5, 2) (23, 2) u' u v v Source: Phil Dadd. Vectors and Matrices: A Primer. www.gamedev.net/articles/programming/math-and-physics/ vectors-and-matrices-a-primer-r1832/ 100. Computer Graphics Refer to Problem 99. The points ( ) ( ) ( ) − − − 3,0 , 1, 2 , 3,1 ,and ( ) 1, 3 are the vertices of a parallelogram ABCD. (a) Find the vertices of a new parallelogram ′ ′ ′ ′ ABCD if ABCD is translated by = 〈 − 〉 v 3, 2. (b) Find the vertices of a new parallelogram ′ ′ ′ ′ ABCD if ABCD is translated by − v 1 2 . 94. Inclined Ramp A 2-pound weight is attached to a 3-pound weight by a rope that passes over an ideal pulley.The smaller weight hangs vertically, while the larger weight sits on a frictionless inclined ramp with angle θ. The rope exerts a tension force T on both weights along the direction of the rope. Find the angle measure for θ that is needed to keep the larger weight from sliding down the ramp. Round your answer to the nearest tenth of a degree. 3 pounds 2 pounds u T T 95. Inclined Ramp A box sitting on a horizontal surface is attached to a second box sitting on an inclined ramp by a rope that passes over an ideal pulley. The rope exerts a tension force T on both weights along the direction of the rope, and the coefficient of friction between the surface and boxes is 0.6 (see Problem 93). If the box on the right weighs 100 pounds and the angle of the ramp is ° 35 , how much must the box on the left weigh for the system to be in static equilibrium? Round your answer to two decimal places. T T F1 F2 W2 FN2 35° FN1 W1 96. Muscle Force Two muscles exert force on a bone at the same point. The first muscle exerts a force of 800 N at a ° 10 angle with the bone. The second muscle exerts a force of 710 N at a ° 35 angle with the bone.What are the direction and magnitude of the resulting force on the bone? 97. Truck Pull At a county fair truck pull, two pickup trucks are attached to the back end of a monster truck as illustrated in the figure. One of the pickups pulls with a force of 2000 pounds, and the other pulls with a force of 3000 pounds. There is an angle of ° 45 between them. With how much force must the monster truck pull in order to remain unmoved? [Hint: Find the resultant force of the two trucks.] 45° 2000 lb 3000 lb
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