Math 132 Spring 2003 Exam 3 Solutions No calculators with a CAS are allowed Be sure your calculator is set for radians not degrees if you do any calculus computations with trig functions Part I Multiple Choice 5 points problem blacken your answers on the answer card 1 A curve is given by B cos 0 1 where is some positive C sin constant The length of the curve is 3 What is the value of A 1 F 1 B G 21 D 1 I 1 C 1 H 1 E 2 3 Method I We have B cos and C sin If we square both sides of each equation and add we get B C cos sin So the point B C moves along the circumference of a circle or radius centered at 1 For 0 1 the point travels over a quarter of the circumference and that length 1 1 Therefore 1 which gives 1 C B 1 Method II We have B sin and cos Since 0 for 0 the B coordinate is always decreasing so a point moving along a curve with those parametric equations moves along the path only once Therefore total distance traveled 1 1 length of curve B C sin cos 1 1 Therefore 1 or 1 2 A finite region in the first quadrant is bounded by the line C B and the curve C B What is the volume of the solid obtained by revolving the region around the Baxis A 1 F 1 B G 1 1 C H 1 1 D 1 I 1 E 1 J 1 The region is pictured below Using the washer method the volume Z 1 outer radius inner radius B 1 B B B 1 B B B 1 B B l 1 1 3 A spring is stretched 2 ft beyond its natural length and this requires 6 ft lbs of work How much additional work would be required to stretch the spring an additional foot A 3 ft lbs F 5 5 ft lbs B 3 5 ft lbs G 6 ft lbs C 4 ft lbs H 6 5 ft lbs D 4 5 ft lbs I 7 ft lbs E 5 ft lbs J 7 5 ft lbs By Hooke s Law the amount of force J B needed for B feet of stretch beyond the natural length of the spring is J B 5B The work done in stretching 2 feet beyond the natural length is 6 ft lbs 5B B 5 B l 5 Therefore 5 The amount of work done in stretching from 2 feet beyond natural length to 3 feet beyond natural length is J B B B B B l ft lbs 4 What is the length of the graph of C B where B A B C D E 9 F G H I 2 J 4 C Method I P B B B B B B Letting B B gives B B l C Method II Let B C Then P B Then the integral works out just as above 5 A tank 10 ft tall has the shape of a circular cylinder its radius is 5 ft It contains water to a depth of 2 ft How much work is done pumping the water up over the top edge of the tank Water weighs 62 5 lbs ft round your answer to the nearest integer All answers are given in ft lbs A 148 742 F 83 753 B 101 257 G 79 841 C 98 175 H 77 981 D 95 332 I 75 369 E 88 357 J 73 224 Let the C axis C be the vertical axis of the cylindrical tank At height C the horizontal cross section of the liquid tank is a circle whose area E C 1 1 ft For C we think of a thin slice of the liquid at height C with thickness say C The slice has volume E C C ft and the liquid in that slice weighs E C C 1 C lbs That slice must be lifted approximately a distance C to get its liquid out of the tank The total work is therefore 1 C C 1 C C l ft lbs 6 The base of a certain solid is pictured Cross sections of the solid perpendicular to the C axis and to the base are quarter circles What is the volume of the solid A 1 F 2 B 1 G 1 C 1 H 41 D 1 I 51 E 51 J 3 For C the radius pictured of the circular cross section is B C Therefore the area of the cross section at C is 1 radius 1 C 1 C Therefore the solid has volume Z E C C 1 C C 1 C l 1 1 7 A woman on an island teeming with hungry mosquitos is being bitten at random times Assume that the random variable time between bites has an exponential density function If the average time between bites is 1 minute what is the probability that the time between bites is two minutes or more Round your answer to 4 decimal places A 0 0031 F 0 2234 B 0 0132 G 0 2437 C 0 1353 H 0 2539 D 0 1478 I 0 2754 The exponential density function is 0 E 0 1798 J 0 2971 where 0 lim lB Therefore T B 8 For the same woman what is the median time between bites All answers are given in minutes and rounded to 4 decimal places A 0 0042 F 0 6931 B 0 1397 G 0 8734 C 0 1459 H 1 2234 D 0 3452 I 1 3457 The median time 7 is that time for which T E 0 5512 J 1 5243 7 T 7 Therefore 7 0 l7 7 7 Therefore e 7 so 7 ln 8 ln 8 8 8 ln 8 9 Suppose all are positive constants Find lim if it exists A F B G C H Consider the function 0 B D I B ln B B ln B use L Hospital s Rule to write lim 0 B Since lim 0 B B B 8 ln 8 lim 8 8 ln 8 E J does not exist B ln B B ln B lim B B B Since lim 0 B we can B B also 10 For what B s does the series B B B 8 8 1 B 8 8 converge A 1 B 1 E B I B only B 1 B 1 C 2 B 2 F B G B J for no values of B D 2 B 2 H B The series is a geometric series with ratio B Therefore it converges when l l l B l and diverges when l l l B l 1 This means the series converges exactly when B 11 Suppose a random variable has probability density function for B or B for B 0 B B B for B What is the mean of A 0 F 1 B 0 2 …
View Full Document
Unlocking...