Question

2) Find the area of the region enclosed by each group of equations. a) y x, y sin x, b) y x2-2x, y= x+ 4 x 3) A particle moves along a straight line with a velocity of 5 cm/s and then accelerates at a rate of a(t) t2+2. What is its displacement after 5 seconds of acceleration (acceleration is in cm/s2) 4) A cylindrical can (radius 10 millimeters) is used to measure rainfall in Stormville. The can is initially empty and rain enters the can during a 60-day period. The height of water in the can is modeled by the function S. where S(t) is measured in millimeters and t is measured in days for 0sts 60. The rate at which the height of water is rising in the can is given by S'(t) 2 sin(0.03t) 1.5. According to the model, what is the height of the water in the can at the end of the 60-day period? 5) The rate (R) at which people enter an auditorium for a rock concert, is modeled by the function R where R(t) auditorium when the doors open (t 0), and the doors close 2 hours later, when the concert begins (t = 1380t2 -675t3, and time (t) is measured in hours. No one is in the are in the auditorium when the concert begins? 2). How many people = 6) A sand bag originally weighing 144 lb was lifted at a constant rate. As it rose, sand leaked out at a constant rate. The sand was half gone by the time the bag had been lifted 18 ft. How much work (ft-lb) bag). was done lifting the sand this far (neglect the weight of the lifting equipment and 7) A rock climber is about to haul up 100 N (about 22.5 lb) of equipment that has been hanging beneath her on 40 m of rope weighing 0.8 N/m. How much work will it take to lift all this? 8) A vertical cylindrical tank measures 30 ft high and 20 ft in diameter. It is full of kerosene weighing 51.2 lb/ft. How much work (ft-lb) does it take to pump the kerosene to the level of the top of the tank (i.e to pump all the kerosene out the top of the tank).

Answer 1

(2) a) The region enclosed by the curves y x, ysin x,x ,x T is shown below: - 2 3T Area x-sin x]dx = +0 CoS X = 2 8 J/2 syx-2x,yx+4 (b) For the intersection of the curves 2-2x x+4 -3x-4=0 (x-4) (x+1)=0x -14 -2x , y = x + 4 is shown below: The region enclosed by the curves y = 4 Cx+4)-(x2-2x)dx = [-x2 + 3x + 4]dx - 20.833 Area

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-1 4 2