210-peters-07sp03-vA

# 210-peters-07sp03-vA - Math 210 — Topics in Finite Math...

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Unformatted text preview: Math 210 — Topics. in Finite Math Spring 2007 — Midterm 3 ' Name: TAi VERSION A Calculators are not allowed. ) b) Show your work for every problem. Answers Without justiﬁcation will not receive credit. ) ) There is a blank page at the end of the exam for if you need it. Score Problem Peints 1. (20 points) Consider the following system of linear equations. that depend on the constant a: 2x—6y=6 3x+ay=10. Determine for which (if any) values of a this system has 0, 1 resp. inﬁnitely many solutions. Solve the system in the case when there is a unique solution (your answer'will depend on a). 2. (10 points) Scientist have discovered a new particle. It can exist in four different states and it can switch states very quickly. If the particle is in a given state now, then there is an 70% chance that it is in the same state after one nano-second, and a 10% chance that it has switched to each of the other three states. Give the transition diagram and transition matrix of this Markov chain. 3. (25 points) The trail mix of the Royal Norwegian marines consists of peanuts and raisins. Each marine must carry 300—600 grams of trail mix on a practice mission. One gram of raisins contains 3 calories and costs 4 cents, while one gram of peanuts contains 6 calories and costs 5 cents. The marine guidelines state that raisins must make up at least one~third of the trail mix and peanuts must make up at least half of the trail mix. (a) Sketch the feasible set. Compute the coordinates of each corner point. (b) Determine when the cost of the trail mix is minimal. [Give your answer in terms of raisins and peanuts, not m and (0) Determine when the calories are maximal. [Again, give your answer in terms of raisins and peanuts] 4. (20 points) Draw the feasible set for the constraints m 2 0, y _>_ 0 and y — 2:5 3 1. For What value of the constant a does there exist a maximum for the objective function y —- am? For What values of a does there exist a minimum? 5. (a) (10 points) Is the Markov chain with the following transition matrix regular? 11 ——0 Si; 12? 505 (b) (15 points) The initial state vector is X0 = [.36 .49 .15], estimate XLOOQOOO. ...
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210-peters-07sp03-vA - Math 210 — Topics in Finite Math...

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