MATH 132 EXAM III solutions FALL 2010 1 An aquarium 6 ft long 1 ft wide and 4 ft high is full of water weighing 62 5 lb ft Find the work needed in ft lbs to pump half the water out of the aquarium A 749 B 749 5 C 750 D 750 5 E 751 F 751 5 G 752 H 752 5 I 753 solution Let the bottom be C 1 and the top C 4 For each C the cross section E C so the weight of a layer is 6 62 5 375 Half the water means 2 C 4 and the distance lifted is C Then work done is given by C C 0 6 G 2 A chain 10 ft long uniformly weighing 32 lb is lying on the ground Find the work in ft lbs required to raise the chain up vertically so that the bottom of the chain will be 2 ft above the ground Note The top will end up 12 feet above the ground A 100 B 150 C 314 D 512 E 1664 F 1600 G 384 H 320 I 224 J 160 solution If C is the ground then we want to lift the top of the chain to C The weight is 32 10 3 2 lb ft For C when the top is lifted to level y the weight is 3 2 y lb For 1 C 2 the weight is always 32 lb So work is C C 0 6 M 3 Suppose the waiting time for a customer s call to be answered is related to a probability density function which has an average waiting time of 3 minutes Find the probability that a customer waits more than 6 minutes A e F e G e H e I e J e K e L e M N e solution T M 4 Find the mean for the probability density function f B A 1 B C 2 12x B H 5 I 3 5 30 B 9 2 A3 J 8 K 3 8 L 58 M 3 10 N 7 10 solution B 0 B B B B B B B B I 2 B C 5 Find the solution of the differential equation B C with initial condition y 0 1 B B B B A e F G H I B J B K B L B M B solution C C B B C B G Given y 0 1 we get 1 G then G Then we have C B So we get y B It can t be y B since y 0 1 H 6 How many years will it take an investment to double if the interest rate is 6 compounded continuously A 2 B 4 C 6 D 8 E 10 F 12 G 14 H 16 I 18 J 20 solution E E Solve E E E E 68 Need years F 7 A tank contains 20 lb of salt dissolved in 6000 L of water Brine that contains 0 01 pounds of salt per liter enters the tank at the rate of 50 L min The solution is kept thoroughly mixed and drains out of the tank at the rate of 50 L min How much salt remains in the tank after 2 hours Find the answer to 2 decimals A 42 84 B 43 52 C 44 68 D 45 28 E 46 34 F 47 12 G 48 62 H 49 74 I 50 26 J 51 64 Note 2 hours is 120 minutes C C C solution C This is seperable so we get C 68l Cl G l Cl G C From y 0 20 we get l Cl Since the right side is never zero and C is sometime positive we get C 120 C C H G 8 Find the values of which would make the function y e a solution to the differential equation yww Cw C E F G H I J K L M N solution y e y w e y ww e Then e So we get solution yww Cw C M 9 A bacteria culture starts with 500 bacteria and grows with a constant relative growth rate After 1 hour there are 1000 bacteria In how many hours from the start will the population be 13 500 Find the answer to 2 decimals A 1 25 B 2 C 2 50 D 3 25 E 4 50 F 4 75 G 5 25 H 5 75 I 6 5 J 7 25 solution We have E 5 and E So 5 5 68 Then E Now solve for t given 13 500 68 68 J 10 Determine whether the sequence 8 8 converges or 8 diverges If it converges then find the limit A 0 B G H I J K L M N 3 1 solution 8 8 8 lim 8 8 8 lim H 11 Find the sum of the series A F G H I J K if it converges L M N 3 1 solution This is a Geometric Series with So it converges to H 12 Which of the following 3 series I 8 2 8 68 8 II 8 2 68 8 8 is are convergent 8 III 38 8 8 8 A I B II C III D I II E I III F II III G all solution H none 8 68 8 diverges by integral test since 68 8 8 diverges by Comparison Test since 68 8 8 8 38 8 converges by Comparison Test since 8 8 38 8 8 8 8 8 2 8 2 8 G B 68 B B 68 68 B l p series with p 1 p series with p 1 13 Suppose the convergent series 8 8 s Using the remainder estimate for the integral test find the smallest integer n for which the remainder R8 That is that s s8 l 0 1 where s8 8 E F G H I J K L M N 0 1 solution V8 8 B B 8 The solution comes from 8 8 L 8 14 Find s the 30 2 partial sum A F G H I for the series J K 8 8 8 L M N solution 8 8 S equal to 8 8 So we get J 15 Among the following 4 series find all those that are convergent but not absolutely convergent I 8 MM 8 8 8 8 8 MMM 8 8 8 MZ 8 8 8 8 A I B II C III D IV E I II F I III G III IV H I II IV I II III IV J all solution I convergent 8 8 8 8 is convergent by Alternating Series test but is not absolutely is divergent p series with p 1 Like the above II 8 p series with p III 8 8 IV 8 8 8 8 8 8 8 8 is convergent but not absolutely since 8 is absolutely convergent since 8 8 1 8 is p series with …
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