Math 10170, Exam 2 April 25, 2014 The Honor Code is in effect for this examination. All work is to be your own. You may use your Calculator. The exam lasts for 50 minutes. Be sure that your name is on every page in case pages become detached. Be sure that you have all 12 pages of the test. PLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. (a) (b) (c) (d) (e) 2. (a) (b) (c) (d) (e)... 3. (a) (b) (c) (d) (e) 4. (a) (b) (c) (d) (e)... 5. (a) (b) (c) (d) (e) 6. (a) (b) (c) (d) (e)... 7. (a) (b) (c) (d) (e) Please do NOT write in this box. Multiple Choice 8. 9. 10. 11. Total
Multiple Choice 1.(6 pts.) Roger and Connor are college roommates. They have neglected to discuss a cleaning schedule for their room. Both roommates have two possible strategies at the end of each week; they either tidy up the room (T) or not tidy up at all (NT). For both roommates the most preferred outcome is that they do not tidy the room but their roommate does. The least preferred outcome for each player is that they tidy the room and their roommate does not. Both roommates prefer the situation where both tidy to the one where neither roommate tidies the room. Use ranking of preferences, 1, 2, 3, 4, as pay-off s for each player where a payoff of 4 is given to the most preferred outcome and a payoff of 1 to the least preferred. Which of the following gives the pay-off matrix for this game? (a) Connor T NT R o T (2, 2) (1, 4) g e r NT (4, 1) (3, 3) (b) Connor T NT R o T (4, 4) (2, 3) g e r NT (3, 2) (1, 1) (c) Connor T NT R o T (3, 3) (1, 4) g e r NT (4, 1) (2, 2) (d) Connor T NT R o T (4, 4) (1, 3) g e r NT (3, 1) (2, 2) (e) Connor T NT R o T (3, 3) (4, 1) g e r NT (1, 4) (2, 2) 2
2.(6 pts.) Consider the following payoff matrix for a two person simultaneous move game: Column Player Row Player R2 (-1, 1) (2, 3) (2, 5) Which of the following shows the equilibrium points (circled) and the reduced payoff matrix: (a) R2 (-1, 1) (2, 3) (2, 5) Reduced (b) R2 (-1, 1) (2, 3) (2, 5) Reduced C2 R3 (3, 3) (c) R2 (-1, 1) (2, 3) (2, 5) Reduced (d) R2 (-1, 1) (2, 3) (2, 5) Reduced C2 R3 (3, 3) (e) none of the above 3
3.(6 pts.) In a baseball league, a pitcher, Colin, has two preferred pitches, a fastball (FB) and a curve (C). A batter, Robert, can anticipate a particular type of pitch or not anticipate any pitch at all. The payoff matrix below shows the estimated batting average for Robert for each situation. Colin FB C FB 0.3 0.25 Robert C 0.2 0.3 None 0.2 0.25 Which of the following gives the expected batting average for Robert when the players play the following mixed strategies: Colin pitches his fastball 25% of the time, his curve 75 % of the time. Robert anticipates the fastball 60% of the time, the curve 20% of the time and neither 20% of the time. (a) 0.23 (b) 0.35 (c) 0.235 (d) 0.26 (e) 0.361 4
4.(6 pts.) Two local snooker players Rose and Cathy compete each weekend at either of two local competitions in Goshen (G) or South Bend (SB) each of which offers prize money. Their expected net gain at each tournament depends on a number of factors including cost of travel, prizes offered, entry costs, home advantage, and whether the other player has entered the tournament or not. The payoff matrix below shows the average weekly payoff (in dollars) for each player depending on which tournament they have entered and which tournament the other player has entered. Rose Cathy G SB G (150, 100) (200, 250) SB (175, 210) (150, 200) If Cathy attends the Goshen Tournament (G) 40% of the time and the South Bend Tournament (SB) 60% of the time and Rose attends the Goshen Tournament (G) 60% of the time and the South Bend Tournament (SB) 40% of the time, what is Cathy s expected weekly payoff? (a) $172 (b) $195.6 (c) $167 (d) $187.6 (e) $427.6 5
5.(6 pts.) A jogger, jogging backwards and forwards on a straight path as shown below recorded the following data. The data shows the position (s(t) kilometers) at time t and instantaneous velocity (v(t) km/min) at each 5 minute time (t) interval after the beginning of the jog. ( The position is calculated with respect to a superimposed axis with the origin at the start of the path and with the positive direction pointing towards the end of the path.) t (min.) 0 5 10 15 20 25 30 35 40 45 s(t) (km) 0.5 1.3 1.97 1.07 2.07 2.8 1.8 1 0 v(t)(km/min) 0 0.11 0.16 0.22 0.16 0.2 0.15 0.21 0.20 0 Start H0 kml end H3kmL Calculate the average velocity and average acceleration on the time interval from t = 30 to t = 40 minutes. (a) Average velocity = 0.18 km/min., Average acceleration = 0.005 km/min./min. (b) Average velocity = 1.8 km/min., Average acceleration = 0.05 km/min./min. (c) Average velocity = 0.025 km/min., Average acceleration = 0.05 km/min./min. (d) Average velocity = 0.18 km/min., Average acceleration = 0.005 km/min./min. (e) Average velocity = 0.025 km/min., Average acceleration = 0.005 km/min./min. 6
6.(6 pts.) A ball is thrown directly upwards from a height of 4 meters above ground at a speed of 30 meters per second. The height of the ball t seconds after it is thrown is given by h(t) = 4 + 30t 4.9t 2. What is the maximum height achieved by the ball? (a) Approximately 8 m. (b) Approximately 49.92 m. (c) Approximately 28.76 m. (d) Approximately 57.31 m. (e) Approximately 34 m. 7
7.(6 pts.) The following graphs show the position, velocity and acceleration of a runner, running (forwards and backwards) along a straight path with a superimposed axis over a 3 hour period. Position miles 15 Velocity miles hr. 20 10 10 0.5 1.0 1.5 2.0 2.5 3.0 5 10 0.5 1.0 1.5 2.0 2.5 3.0 Acceleration miles hr. hr. 40 20 0.5 1.0 1.5 2.0 2.5 3.0 20 40 Consider the following time intervals: which of the following is true: A : [0.5, 1], B : [2, 2.3], C : [2.5, 3] (a) (b) (c) (d) (e) The runner is speeding up on intervals A and B and slowing down on interval C The runner is speeding up on intervals A and C and slowing down on interval B The runner is slowing down on all three intervals The runner is speeding up on all three intervals The runner is speeding up on interval C and slowing down on intervals A and B 8
Partial Credit You must show your work on the partial credit problems to receive credit! 8.(15 pts.) Two high school football players Robert and Chris can play effectively in a number of positions on the team. Robert who plays for the Eagles can play as running back (RB), quarterback(qb) or wide receiver(wr). Chris who plays for the Hawks can play tackle (T), middle linebacker (ML), or cornerback (C). Both teams are about to play in the regional championship game. The matrix below shows the estimated expected yardage gained per play for the Eagles (Robert s team) for each choice of positions for Robert and Chris. This can be considered as a zero sum game where yards gained by one team corresponds to yards lost by the other. Chris T ML C RB 5-5 8 Robert QB -5 15 20 WR 10 5 7 (a) Find the reduced payoff matrix by eliminating dominated strategies. (b) Find the optimal strategy for both players using the reduced matrix. (c) What is the expected pay-off for The Eagles (Robert s team) if both players play optimal strategies? 9
9.(15 pts.) A diver diving off a 10 meter(high) board has an initial vertical velocity of 8 m/s. The diver s height t seconds after they jump is approximately given by h(t) = 10 + 8t 4.8t 2 meters. (a) How many seconds elapse after the diver jumps before the diver hits the water? (b) Give a formula for the velocity of the diver at time t (c) What is the vertical velocity of the diver when they hit the water? 10
10.(20 pts.) This problem appears as Problem 1 on the take home part of the exam. You may use this page for rough work. 11
11.(8 pts.) This problem appears as Problem 2 on the take home part of the exam. You may use this page for rough work. 12
Math 10170, Exam 2 April 25, 2014 ANSWERS The Honor Code is in effect for this examination. All work is to be your own. You may use your Calculator. The exam lasts for 50 minutes. Be sure that your name is on every page in case pages become detached. Be sure that you have all 12 pages of the test. PLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. (a) (b) ( ) (d) (e) 2. (a) ( ) (c) (d) (e)... 3. (a) (b) (c) ( ) (e) 4. (a) ( ) (c) (d) (e)... 5. ( ) (b) (c) (d) (e) 6. (a) ( ) (c) (d) (e)... 7. ( ) (b) (c) (d) (e) Please do NOT write in this box. Multiple Choice 8. 9. 10. 11. Total