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1 Section 1.1
```Math 1070Q
Final Review
Name:
August 3, 2015
1
Section 1.1
1. A company finds it can sell 10 items at a price of \$8 each and sells 15 items at a price of \$6
each. Find the demand equation.
2. A contractor needs to rent a wood chipper for a day for \$150 plus \$10 per hour. Find the cost
function.
3. Assume that the linear cost and revenue models apply. An items sells for \$10. If fixed costs
are \$2000 and profits are \$7000 when 1000 items are made, find the cost equation.
2
Section 1.2
4. If the demand equation is D(x) = −x + 6 and the supply equation is S(x) = x + 3, find the
equilibrium point.
5. A firm has a weekly fixed cost of \$40,000 associated with the manufacturing of purses that cost
\$15 per purse to produce. The firm sells all the purses it produces at \$35 per purse. Find the
cost, revenue and profit equations. Find the break-even quantity.
6. You have to options for living: rent an apartment for \$600 per month, or buy a house where
your down-payment is \$20,000 and monthly payment of \$200. Assuming all other expenses are
equal, which option is cheaper to live in for 5 years?
3
Section 3.1
7. A dealer has 7600 pounds of peanuts, 5800 pounds of almonds and 3000 pounds of cashews to
be used to make two mixtures. The first mixtures whosales for \$2 per pound and consists of
60% peanuts, 30% almonds and 10% cashews. The second mixture wholesales for \$3 per pound
and consists of 20% peanuts, 50% almonds and 30% cashews. Write a linear programming
describing how to answer the question “How many pounds of each mixture should the dealer
make in order to maximize revenue?”.
4
Section 3.2
8. Graph the feasible region and find all corner points of
(a)
−x + y ≤ 4
x+y ≥2
x, y ≥ 0
(b)
y≤x
y ≥x−2
y≤4
y≤0
(c) The system from question 7.
5
Section 3.3
9. Find that maximum and minimum values of z = 2x − y given the feasible region
(a)
(4,6)
•
y
(1,5)
•
•(5,3)
•
•
(1,1)
(4,1)
x
(b) 8.8(a)
(c) 8.8(b)
6
Section 4.1
10. Use the following Venn Diagram to indicate where the sets are located
A
I
IV
B
II
I
VII
C
(a) A ∩ B ∩ C
VI
III
VIII
(b) A ∩ B c ∩ C c
(c) A ∩ B ∩ C c
(d) Ac ∩ B c ∩ C c
(e) A ∪ B ∪ C
(f) (A ∪ B) ∩ C c
(g) (A ∪ B ∪ C)c ∩ A
11. Let F = {x|x is female}, M = {x|x is male}, S = {x|x is from Stratford}, N = {x|x is from New Haven}
and B = {x|x plays basketball}. Write an expression to describe
(a) A female from Stratford who plays basketball
(b) A male who doesn’t play basketball and doesn’t live in New Haven
(c) A female from neither Stratford nor New Haven who doesn’t play basketball
(d) A male from Stratford or New Haven
12. Draw a Venn Diagram that shows the property (A ∪ B)c = Ac ∩ B c
7
Section 4.2
13. If n(B) = 100, n(A ∪ B) = 175 and n(A ∩ B) = 40, what is n(A)?
14. In a survey of 500 people, 200 own a dog but not a cat, 150 own a cat but not a dog, and 100
own neither a cat nor dog. How many own both a cat and a dog? How many own a dog?
15. A survey of 500 adults found that 190 play golf, 200 ski, 95 play tennis, 100 play golf but do
not ski or play tennis, 120 ski but no not play golf or tennis, and 40 did all 3.
(a) How many play golf and tennis but not ski?
(b) How many play tennis but do not play golf or ski?
(c) How many play at least one of the three sports?
8
Section 4.3
16. A coin is flipped and a die is rolled.
(a) What is the sample space?
(b) Write a set that indicates the event “Heads and even”
(c) Write a set that is mutually exclusive from set in (b)
17. An urn holds 10 identical balls except that 1 is white, 4 are black, and 5 are red. An experiment
consists of selecting two balls in succession without putting the first ball back.
(a) What is the sample space?
(b) Write a set that indicates the event“no balls are white”
(c) Write a set that indicates the event “both balls are white”
9
Section 4.4
18. A card is drawn randomly from a standard deck of 52 cards. Find the probabilities of drawing
(a) an ace
(c) a club
(d) a red card
(e) a face card (Jack, Queen or King)
(f) a red 8
19. Over a number of years, the grade distribution in a math course was observed to be
A
25
B
35
C
80
D
40
F
20
What is the empirical probability that a randomly selected student taking this course will
20. An urn holds 10 identical balls except that 1 is white, 4 are black, and 5 are red. An experiment
consists of selecting one ball.
(a) What is the probability of selecting a white ball?
(b) what is the probability of selecting a ball that is not white?
10
Section 4.5
21. A company believes that it has a probability of 0.40 of receiving a contract. What is the odds
that it will?
22. The odds of a marriage ending in divorce is 1:1. What is the probability of getting a divorce?
23. An inspection of computers reveals that 2% of the monitors are defective, 3% of the keyboards
are defective, and 1% of the computers have both defects. Find the probability that a computer
has one OR the other defective.
11
Section 4.6
24. A coin is flipped three times. What is probability that heads occurs three times if it is known
that heads occurs at least once?
25. An urn contains five white, three red, and two blue balls. Two balls are drawn randomly. What
is the probability that one is white and one is red if the balls are drawn
(a) without replacement?
(b) with replacement after each draw?
26. A company sells machine to two firms. In 40% of the years it makes a sale to the first firm,
30% to the second firm, and 10% to both. Are the two events “a sale to the first firm” and “a
sale to the second firm” independent?
12
Section 5.1
27. A restaurant offers 4 types of salads, 10 main dishes, and 5 desserts. How many different
complete meals are there?
28. In how many ways can a five member basketball team line up for a picture?
29. In how many ways can a horse race of 10 horses have a different trifecta? (1st, 2nd and 3rd
place finishers).
30. How many different licence plates containing 6 places that are either digits or letters without
repeats? Ex: AJ73K1.
13
Section 5.2
31. Find the number of ways to get the following hands in a poker game (where you have 5 cards
out of the possible 52). I have given an example of each hand, but we are looking for the
number of hands of that format in general, not the specific hand given in the example.
(a) Straight flush (Ex: 3,4,5,6,7 all of clubs). H int: There are 10 possible straights
(b) Four of a kind (Ex: AAAAK)
(c) Full House (Ex: AAAKK)
(d) Flush (Ex: A,2,5,7,10 of clubs)
(e) Two Pair (Ex: AAKKQ)
(f) One pair (Ex: AAKQ10)
14
Section 5.3
32. Find the probability of each of the hands from problem 11
33. What is the probability that at least two members of the 434-member US House of Representatives have their birthdays on the same day?
15
Section 5.4
34. Find the probability of exactly 3 success in 6 repeated Bernoulli trials with the probability of
success is 0.5.
35. You roll a die and win if the number showing is a 1 or a 2. If you roll the die 8 times, find the
probability of
(a) winning exactly 6 times.
(b) never winning.
(c) winning at most 6 times.
16
Section 6.1
36. A pair of dice are thrown. Let X be the random variable given by the absolute value of the
difference in the two numbers.
(a) Find the probability density
(b) Draw a histogram
(c) What is the probability of the two numbers being identical?
37. The probability distribution of the random variable X is given by the following table. Complete
the following
x
-2
-1
0
1
2
3
P (X = x) 0.2 0.15 0.05 0.35 0.15 0.10
(a) Draw a histogram
(b) Find P (X = 0)
(c) Find P (X ≤ 0)
(d) Find P (−1 < X ≤ 4)
(e) Find P (X ≥ 1)
17
Section 6.2
38. Find the mean, median and mode of {3, 5, 8, 9, 12, 14}
39. If the mean for a binomial distribution is 1.2 and there were six trials, find p.
40. Find the expected value of the random variable X having the probability distribution in problem
17.
41. In American roulette (18 red, 18 black, 2 green) consider a \$1 bet split on 0 and 00. If either
numbers hit, you have \$18 returned to you (a profit of \$17). If neither of those numbers hit,
you lose your \$1. Find the expected return on this bet.
18
Section 6.3
42. Find the variance and standard deviation of the random variable X having the probability
distribution in problem 17.
43. In a recent golf tournament the top five
of the tournament
B
P
S
M
A
finishers had the following scores for the four rounds
67
69
67
70
70
71
67
69
66
65
67
70
67
72
73
68
71
74
70
70
Find the mean, variance and standard deviation for each player. Which one had the lowest
average? Which one was the most consistent? Least consistent?
44. A probability distribution has a mean of 90 and a standard deviation of 3. Use Chebyshev’s
inequality to find the minimum probability that an outcome is between 60 and 120.
19
Section 6.4
45. Evaluate the following
(a) P (Z ≤ −0.3)
(b) P (1.0 ≤ Z ≤ 1.5)
(c) P (Z ≥ −1.42)
(d) P (x ≤ 50), µ = 38, σ = 8
(e) P (100 ≤ x ≤ 150), µ = 20, σ = 100
46. Find the area under the standard normal curve on the interval [-2,2].
20
Section F.1
47. Find the simple interest and future value if P = \$1000, r = 8% (r is the annual rate) and the
loan is taken out for 4 months.
48. A principal of \$2000 earns 6% per year simple interest. How long will it take for the future
value to become \$2300?
21
Section F.2
49. Find how much is in an account that has a principal of \$1000 with annual rate of 10% compounded monthly after 3 years.
50. Find the effective yield given annual rate of 8% compounded semiannually.
51. Find the present value needed to have \$10,000 in a savings account that receives 9% interest
compounded weekly after 20 years.
22
Section F.3
52. An individual earns an extra \$2000 each year and places this money at the end of each year
into an IRA which has an annual interest rate of 12% compounded annually. How much is in
the account at the end of 40 years?
53. A couple needs \$20,000 at the end of five years for a down payment on a house. How much
should they place into a savings account earning an annual rate of 7% compounded monthly
to meet this goal?
23
Section 2.1
54. If

3
−2
A=
0
2



1
1
4

4
6
,B =  3

−3 7  ,
3
8
3 −2
Find
(a) The order of A
(b) the element a12
(c) the element a21
(d) the element b42
(e) A + B
(f) B − A
55. Find the values of a, b, c, d for the matrix equation
24
T 6 4
−2 1
a b
=
+3
0 7
4 x
c d
Section 2.2
56. Find the order of AB and BA if A is order 2 × 3 and B is order 3 × 4.
57. Perform the multiplications if possible
(a)
2 5 2
4 3 1
(b)
2 5
2 1
4 3
(c)
2 1
1 0
2 5
4 3
(d)

 
2
1 3
2
0
1 0 −1
−1 0.5 2
0
(e)



2
1 3
2 −1 −1
0
1 0 −1 4 −2
−1 0.5 2
0
0
1
Material After Exam 2
25
Section 1.3
58. Find the solution of the following systems (if it exists)
(a)
2x + 4y = 8
2x − 4y = 0
(b)
x+y+z =1
2x − y + z = 2
3x + 2y + 5z = 0
26
Section 2.3
59. Find the inverse of
(a)
1 3
A=
1 4
(b)


1 0 1
A = 0 2 1
1 0 2
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