Review Problems: Math 131 – Final
1.
2.
3.
The RideEm Bicycles Factory can produce 100 bicycles in a day at a total cost of $10,500, and it
can produce 120 bicycles in a day at a total coat of $11,000. What are the company’s daily fixed
costs, and what is the marginal cost?
You can sell 90 Chia pets each week if they are marked at $1/Chia but only 30 each week if they
are marked at $2/Chia. Your Chia supplier is prepared to sell you 20 Chias each weekif they are
marked at $1/Chia and 100 each week if they are marked at $2/Chia.
a. Write down the associated linear demand and supply functions.
b. At what price should the Chia be marked so that there is neither a surplus nor a shortage
of Chias.
Sketch the following inequalities. Also, find the coordinates of the corner points of the feasible
solution set.
a.
x  3y  9
2x  y  6
x 0
y 0
b.
2x  3y  12
x y6
4.
Maximize C = x + 2y
Subject to:
30x  20y  600
0.1x  0.4y  4
0.2x  0.3y  4.5
x 0
y 0
5.
Minimize C = .2x + .3y
Subject to:
0.2x  0.1y  1
10x  10y  80
.15x  .3y  1.5
x 0
y 0
6.
You manage an ice cream factory that makes two flavors: Creamy Vanilla and Continental Mocha.
In each quart of Creamy Vanilla go 2 eggs and 3 cups of cream. Into each quart of Continental
Mocha go 1 egg and 3 cups of cream. You have in stock 500 eggs and 900 cups of cream. You
make a profit of $3/quart of Creamy Vanilla and $2/quart of Continental Mocha. How many quarts
of each flavor should you make in order to earn the largest profit?
7.
Enormous State University’s Business school is buying computers. The school has two models
from which to choose, the Pomegranate and the iZac. Each Pomegranate comes with 400
megabytes of memory and 80 gigabytes of disk space; each iZac has 300 megabytes of memory
and 100 gigabytes of disk space. For reasons related to its accreditation, the school would like to
say that it has a total of at least 48,000 megabytes of memory and at least 12,800 gigabytes of disk
space. If the Pomegranate and iZac cost $2000 each, how many of each should the school buy to
keep the cost as low as possible.
8.
Each serving of Gerber Mixed Cereal for Baby contains 60 calories and 11 grams of
carbohydrates. Each serving of Gerber Mango Tropical Fruit Desert contains 80 calories and 21
grams of carbohydrates. If the cereal costs 30 cents per serving and the desert costs 50 cents per
serving and you want to provide your child with at least 140 calories and at least 32 grams of
carbohydrates, how can you do so at the least cost?
Determine the monthly payment on $20,000, 6-year loan borrowed at 8% annual interest.
9.
10. Meg’s pension plan is an annuity with a guaranteed return of 10% interest/year (compounded
monthly). She would like to retire with a pension of $2000 per month for 20 years. If she works 40
years before retiring, how much money must she and her employer deposit each quarter?
11. You want to buy a 10-year zero-coupon bond with a maturity value of $20,000 and a yield of 6%
annually. How much will you pay?
12. The simple interest on a $1000 loan at 8% amounted to $640.00. When did the loan mature?
13. At auction, 1-year - $1,000 treasury bills were sold at a discount of 4.25%. What was the annual
yield?
14. A bag contains 3 red and 3 green marbles. Suzy grabs four marbles at random. X is the random
variable “the number of red marbles Suzy has in her hand”. Find the expected value of X. Also
find the standard deviation,  , of the random variable X.
[Hints: Complete the following table.]
x
P(X  x)
x P(X  x)
X  
(X  )2
(X  )2 P(X  x)
15. Compute the mean, median and mode of the given data sample:
S = {1, 2, 2, 2, 4, 5, -1, -1, -2}.
Mean =
Median =
Mode =
16. You need to take out a loan of $25,000 to buy a car. You have two options: One bank is offering a
10% loan for 6 years and another is offering 8% for 4 years. Which will have the lower monthly
payments? On which will you end up paying more interest total? You can afford to pay back
$390.00 per month. Which loan should you get?
17. Twenty darts are thrown at a dartboard. The probability of hitting a bull’s-eye is .2. Let X be the
number of bull’s-eye hit. Calculate the expected value, the variance, and the standard deviation of
the given random variable X.
18. Compute the variance and the standard deviation of the data sample {2, 5, 7, -1, -2, -2}.
19. Given: S = {a, b, c, d, p, q, r, s, w, x, y, z}; A = { a, b, c, d, x, y, z}; B = {b, d, w, x, y, z, p, r, s},
and C = {b, c, x, y, q, r}. Find the following sets:
a.
b.
c.
A B 
A  B  C  
A  B   



20. The local diner offers a meal combination consisting of an appetizer, a soup, a main course, and a
dessert. There are five appetizers, two soups, four main courses, and ten desserts. You diet restricts
you to choosing between a dessert and an appetizer. (You can not have both.) Given this
restriction, how many three course meals are possible?
21.
a.
b.
c.
How many 6 letter sequences are possible that use the letters J, A, C, K, I, E at most once
each?
How many 4 letter sequences are possible that use the letters J, A, C, K, I, E at most once
each?
How many sets of 4 letter are possible that use the letters J, A, C, K, I, E at most once
each?
22. If 20 business people have a meeting and each pair exchanges business cards, how many
business cards, total, get exchanged?
23. The following table shows the performance of a selection of 100 stocks after one year. (Take S to
be the set of all stocks represented in this table.)
Companies
Pharmaceuticals
Electronic (E)
Internet (I)
Total
(P)
Increased (V)
10
5
15
30
Unchanged (N) 30
0
10
40
Decreased (D)
10
5
15
30
Total
50
10
40
100
Based on this table, answer the following questions. Compute nP  N  . Calculate
nV  I 
. What
n(I)
does the answer mean?
24. A binary digit, or “bit” is either 0 or 1. A nybble is a four-bit sequence. [An example of a nybble
is 0001]. How many different nybbles containing a single 1 are possible?
25. Your international diplomacy trip requires stops in Thailand, Singapore, Hong Kong, and Bali.
How many possible itineraries are there? How many possible itineraries are there in which the last
stop is not Bali.
26. Two distinguishable dice are rolled. Events E, F, and G are defined below.
E = “Both numbers add to 11”
F = “Both numbers are odd”
G = “One number is even and the other number is odd”
Complete the probability distribution table for these events.
E
F
G
Probability,
P
27. You are given n(A B) 15;
the following Venn-diagram.
n(B C)  27; n(C)  42; n(S)  62 . Find j, k, m, and n in
A
B
k=
j=
5
m=
17
11
C
10
n=
S
28.
The table shows
17.
a selection of well-known companies
listed in the S&P 500 together with their 1999 second
quarter earnings. Suppose that from this list you had
selected three companies at random. What is the
probability that
a. All three stocks in your selection had negative
earnings.
b. All three stocks in your selection had positive
earnings.
Company
America online
Compaq Computers
Humana
Gillette
National Semi
Cabletron Systems
Cummins Engine
2nd Quarter
Earnings
- 25.2%
-25.2%
-25.0%
-30.8%
+171.0%
+58.8%
+61.5%
29. In 2001 the probability that a randomly selected online household had cable access was .11, and
the probability that an online household had DSL access was .05. What percentage of online
households had neither cable nor DSL access? (Assume that the probability that an online
household had both kinds of access is negligible.)
30.
A bag contains 4 red, 3 green, 2 yellow, and 1 pink marbles. You grab 4 marbles at random.
a. What is the probability that you would get all red?
b. What is the probability that you would get one pink and at least one yellow?
31. What is the probability that you would get the two yellow one given that you have picked the pink
one?
32. The following pi chart shows the percentage of the population that uses the internet, broken down
by family income, based on a survey taken in August 2000:
< 35,000, User
13%
>35,000,
Nonuser
19%
< 35,000,
Nonuser, 26%
< 35,000, User
42%
(a) What is the probability that a randomly chosen person was an internet user?
(b) Determine the probability that a randomly chosen person was an internet user, given that his
or her family income was at least $35,000.
C P(A  C)  .06
33. Supply the missing quantities:
.1
A
D
__________=______
E
__________=______
.2
B
F __________=______
34. The random variable X has the following probability table:
x
2
4
6
8
10
PX  x 
a.
.2
.3
.3
Assuming that PX  2 is one fifth of PX  10 , find the missing values.
b. Calculate PX  8
35. It rains in Spain an average of once every 10 days, and when it does, hurricanes have 3% chance of
happening in Hartford. When it does not rain in Spain, hurricanes have a 1% chance of happening
in Hartford. What is the probability that it rains in Spain when hurricanes happen in Hartford?
36. According to a study, the probability that a randomly selected teenager watched a rented video at
least once during a week was .75. What is the probability that at least 8 teenagers in a group of ten
watched a rented movie at least once a week?
37. In a large on-the-job training program, 25% of the participants are men and 75% are women. In a
random sample of 5 participants, what is the probability that (a) all of them are men? (b) at least
one of them is a women?
38. Sixty darts are thrown at a dartboard. The probability of hitting a bull’s-eye is .25. Let X be the
number of bull’s-eye hit. Calculate the expected value, the variance, and the standard deviation of
the given random variable X.
39. Following are the hourly worker compensation costs, in U.S. dollars, for five European countries:
{ 17, 21, 24, 17, 21 }. Compute the sample mean and standard deviation.
40. A $5000.00 loan, taken now, with a simple interest of 8% per year, will require a total payment of
$7400.00. When will the loan mature?
41. At auction on January 10, 2000, 1-year T-bills were sold at a discount of 5%. What was the annual
yield?
42. You deposit $10,000 in an account at the Lifelong Trust Savings and Loan that pays 5% interest
compounded quarterly. What is the amount in your account after 5 years?
43. Determine the amount of money, to the nearest dollar, you must invest now at 10% per year
compounded annually, so that you will be a millionaire in 25 years.
44. Your pension plan is an annuity with a guaranteed return of 5% interest / year (compounded
quarterly). You can afford to put $1200 / quarter into the fund, and you will work for 40 years
before retiring. After you retire you will be paid a quarterly pension based on a 25-year payout.
How much will you receive each quarter?
45. Find the mean median and the mode of the following data set
{4, 5, 6, 7, 3, 2, 3, 2, 1, 4, 3, 7}
46. According to a study, the probability that a randomly selected teenager watched a rented video at
least once during a week was .75. What is the probability that at least 8 teenagers in a group of ten
watched a rented movie at least once a week?
47. A manufacturer of a light bulbs chooses bulbs at random from its assembly line for testing. If the
probability of a bulb’s being bad is 0.1, how many bulbs do they need to test before the probability
of finding at least two bad ones rises to more than .5?
48. X is a binomial variable with n = 8 and p = .3. Find the following probabilities.
PX  1
PX  7
PX  7
PX  7
PX  7
PX  8
49. Find the periodic withdrawal for the given annuity: $150,000 at 5%, paid out monthly for 15 years,
leaving 20,000 in the account at the end of 15 years.
n
x
x
i1
n
n
i
; E(X )   
x
N
 x P X  x ;   np; s
i
i
2

i1
n
i  x
i1
n 1
2
;
2 
x
 
2
i
i1
n
;
 r mt
  E (X  )   x i   P X  x i  ;   npq ; FV  PV(1 rt) ; FV  PV 1  ;
 m 
i1
n
2
FV  PV(1 i)  PMT
n




2
2
1 in 1
i
2
1 1 i 
; PV  FV (1 i)  PMT
i
n
n