M230 Test III Fall 1998 Solutions

1. The Feasible set is shaded:

The maximum value of x - 3y is 9 at the point (9, 0). The maximum value of y-3x is 0 at the point (0,0). So the points are not the same.

2.

So,

3. Let

R1-R2

Thus

4. Let
a = # of airplanes produced
b = # of boats produced
c = # of cars produced

Then , and , since it is not possible to produce a negative number of anything. Also the plastic used cannot exceed 10,500 grams, so :

The total wood usage cannot exceed 1500 inches, so :

Finally the steel usage cannot be greater than the 25,500 grams available, so:

The total profit for a given production schedule is:
3a + 5b + 4c ,
so this function needs to be maximized subject to the given six constraints.

5.
(a) Transition diagram:

(b) Transition matrix: (States in order 1-2-3 for rows and columns.)
OR

BONUS

So the probability of state # 2 after 3 years (after starting in state # 2) is . (underlined in ).

OR using the tree diagram below: The probability a person infected now is also infected 3 years from now is:

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