| Finite Math |
Final Exam
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Dec. 13, 2000
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| A Track |
R. Hammack
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| Name: ________________________ |
Score: _________
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(1) Solve the following systems of equations.
(a)
| 3x1 | + | 6x2 | + | 9x3 | + | 6x4 | = | 9 |
| 2x1 | + | 4x2 | + | 6x3 | + | 2x4 | = | 4 |
(b)
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x
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+
|
y
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-
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z
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=
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0
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|
x
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+
|
y
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+
|
z
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=
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2
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|
x
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-
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y
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+
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z
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=
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2
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(2) Find the maximum of the objective function P = x - 10y subject to
the following inequalities
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x + y
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≤ 4 |
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x - y
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≤ 2 |
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x
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≥ 0 |
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y
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≥ -1 |
Notice that this linear programming problem is not in standard form. Thus the simplex method will not work. Solve it by graphing.
(3) Use the simplex method to maximize P = 2x + 3y + z, subject to the following constraints:
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2x
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+
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y
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+
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z
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≤ 2
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4x
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+
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4y
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≤ 4
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||
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3x
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+
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3y
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+
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z
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≤ 9
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x, y, z
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≥ 0
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||||
(4) A hand of 4 cards is dealt off of a standard 52-card deck. How many such hands are possible if:
(a) Order is considered?
(b) Order is not considered?
(5) Four cards are dealt off a well-shuffled deck.
(a) You win $1 if the first card is a club, the second is a diamond, the third is a heart, and the fourth is a spade. Find the probability of your winning.
(b) You win $1 if the four cards are of different suits. Find the probability of your winning.
(6) A fair die is rolled 5 times. Find the probabilities of the following events.
(a) All rolls have an even number of spots.
(b) It is not the case that all rolls have an even number of spots.
(7) Suppose A and B are events. Suppose that P(A) = 0.5, P(B) = 0.6,
and P(A ∩ B) = 0.3.
(a) Find P(A ∪ B).
(b) Are A and B independent? Explain.
(c) Are A and B mutually exclusive? Explain.
(8) Two cards are dealt off a well-shuffled deck.
(a) What is the probability that both cards are black?
(b) What is the probability that either both cards are black or both are aces?
(c) What is the probability that both cards are black and both are aces?
(9) Find the optimal stratigies for each player in the following matrix
games. Determine if each game is fair, and if not, which player has the advantage.
(a)
| [ |
-3
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4
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] |
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-1
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2
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(b)
| [ |
-4
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2
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3
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] |
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2
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-1
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-1
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(10) Imagine that you own a movie rental business with two locations, Location A and Location B. A movie rented at one location may be returned to the other. You notice that each week, 90% of the movies rented at Location A are returned to Location A, while 70% of those rented at Location B are returned to Location B. An inventory taken today suggests that 40% of the movies are at Location A, and the remaining are at Location B.
(a) What percentages of the movies will be at the two locations next week?
(b) What percentages will be at the two locations in the long run?