# Problems 41-60

• 41.   Partial differential equations required. Water flows into a tank at a rate of 1 gallon per second. Water leaves the tank at a rate of 1 gallon per second for each 100 gallons in the tank. The tank is initially empty. How long will it take for the tank to fill with 50 gallons of water? (Answer), (Solution).

• 42.  Probability and algebra requied. A drunk is on one point of an n-gon, such that n is an even number. The drunk moves along the perimeter of the n-gon. Each steps takes him to an adjacent point, and every step is chosen at random. What is the expected number of steps before he arrives at the point directly opposite of his starting point (as a function of n)? (Answer), (Solution).

• 43.   There exists a city surrounded by a circular wall. The wall has two gates, one on the north end and one on the south end. A yellow house exists 3 miles north of the north gate. A blue house exists 9 miles east of the south gate. The yellow and blue houses can barely see other around the side of the wall. What is the radius of the circle formed by the wall? You may visit the The MathServ Calculus Toolkit for help factoring polynomials. (Answer), (Solution).

• 44. Probability and infinite sums requied. You are given an initial prize of one dollar. A coin will be tossed indefinitely until a tails occurs. Every time a heads is tossed your prize will be doubled. When a tails is tossed you walk away with whatever your prize has grown to. For example if you toss three heads and then tails you would get \$1*2*2*2=\$8 dollars. What is the expected return of this game? (Answer), (Solution).

• 45.  Integral calculus and trigonometry required. What is the ratio of the surface area of the state of Colorado to the surface area of the earth? You may assume that the lower and upper lines of latitude are 37 degrees and 41 degrees, and that the east and west lines of longitude are 102 and 109 degrees (the latitude lines I believe are exactly right and the longitude lines are very close and both appear to be off by about the same amount in the same direction). You may also assume the earth is perfectly spherical (which it isn't). (Answer), (Solution).

• 46.   Probability and infinite sums requied. Assuming you flipped a coin until over and over, keeping track of the number of both heads and tails flipped, what is the expected number of flips before the two totals were equal? (Answer), (Solution).

• 47.  Geometry required. What is the height of a pentagon with side length of 1? Answer many not be expressed in trigonometric functions. (Answer), (Solution).

• 48.    Geometry required. What is the volume of a dodecahedron with side (of pentagon) equal to 1? Hint: you may wish to solve problem problem 123 first. (Answer), (Solution).

• 49.  Interest theory requied. This situation actually happened to me at Toyota of Garden Grove, although the numbers were changed for the benefit of simplicity: You are about to pay \$10,000 cash for a new car. The finance manager, wanting to loan you money, asks you what interest rate you could earn on that money if you didn't spend it. You answer 6%, compounded monthly. The finance manager then says that he will give you a loan at 7.5% interest, compounded monthly, with a five year term. He then shows you on his computer that if you invested the \$10,000 on your own you would earn \$3,488.50 in interest in five years. Then he shows you that your payments on the car would be \$200.38 a month, making total interest paid on the car \$2,022.77. You do the math on your own and you realize it is correct. He then argues that you would come out ahead by taking out a loan since you would make more in interest than you would pay. Is this argument valid? (Answer), (Solution).

• 50.  Probability required. The world series consists of up to seven games, the first team to win four wins the series. Team A will play in their home field games 1, 2, and 5, 6, and 7 if necessary. Games 3, 4, and 5 (if necessary) will be played at team B's home field. Assuming each time has a 50% chance of winning every game, which team will is more likely have have the home field advantage the majority of the time? For extra credit what is the expected number of games played in each home field? (Answer), (Solution).

• 51.  Geometry and derivative calculus required. You have a circular piece of paper with radius 1 and wish to form a cone shaped cup. You may cut out any wedge you like from the paper, call the angle of the wedge x. The point of the wedge must be at the center of the circle. After cutting out the wedge you then attach the two straight edges remaining to form a cone. Assuming the paper cone can hold water, what should x be to maximize the water holding capacity of the cone? (Answer), (Solution).

• 52. A woman is chosen at random among all women that have two children. She is asked do you have at least one boy, and she answers 'yes.' What is the probability her other child is a boy? Assume every pregnancy has a 50/50 chance to be a boy or a girl. (Answer), (Solution).

• 53.    Integral calculus and trigonometry required. What is the mean distance between two random points in a unit square?

Note: This problem was on the list of unsolved problems (this list no longer exists) from 5/19/97 to 9/15/97.

• 54. Derivative calculus required. Your job allows you to work any number of hours per week you desire. Your take home pay is proportional to the number of hours worked (no overtime). After subtracing time for sleeping and routine daily tasks you have 80 hours per week left for work and pleasure. You wish to maximize your income multiplied by the amount of pleasure time you have to enjoy it. How many hours per week should you work? (Answer), (Solution).

• 55.     • 56.  Probability required. A baby is added to a hospital nursery. Before the baby was added there were two boys in the nursery and an uncounted number of girls. After the new baby is added a baby is selected at random among all the babys. The selected baby is a boy. What is the probability that the added baby was a girl? (Answer), (Solution).

• 57. No math required but spacial relations helpful. Given an 2n by 2n chessboard, with one corner removed, show that you can cover it completely without overlapping or going off the board using only triominoes of this shape: Rotations are allowed. (Answer & Solution).

• 58. Simple math required. How many subsquares can you form on chessboard with n*n unit squares? Subsquares must be an integer squared. (Answer), (Solution).

• 59.  Algebra required. Show that any prime number other than 2 can be expressed as the difference of two squares, where each square is an integer squared. (Answer).

• 60.   Computer helpful. Two players are each dealt a card face down. Each player may look at his own card. The highest card wins. Cards are valued as in poker with aces being low. The first player may either keep his card or switch with the second player. The second player may keep his card, whether it be his original card or one that the first player gave him after switching, or trade it with the next card on the deck, which is also face down. The loser pays the winner \$1 and if both cards are equal then no money exchanges hands. For the sake of simplicity assume an infinite number of decks. Both players are infinitely logical. At what point should the first player switch? At what point should the second player switch if the first player doesn't switch? What is the expected gain of the first player? (Answer), (Solution).

Michael Shackleford, ASA