Download Math 10 Final Exam Review

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Math 10
Final Exam Review
1. State whether each statement is true or false.
a. The set of days in a month is a well defined set.
b.
{
}c0
c.
4 . 5 e { x | x 6 7Vand3<x<7}
d.
{2,4,6,8,...} n{3,6,9„12,...} = { }
2. Express the set in roster form.
{x|xe7Vand2<x<5}
3. Express the set in set builder form.
{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}
For problems #14-20, use the follovwng information to fill in the Venn diagram and answer the questions.
f/={a, b, c, d, e,f}
A ={a, b, d, e}
B ={a, c, e, f}
C={a, c, d}
4. ^^-dAHB.^s^
5. Find AClB'.
u
A
B
--^ ^
t
\
i r \
\eg)
\
|
6. Find . 4 U ( s n c ' ) . ^ a c J
\
c
~\
7. Find n{A (15
8. Find A-B.
^
^1^4^
9. Find the distinct number of subsets of U.
10. Find the distinct number of proper subsets of f/.
/
^
11. Let set A = {Sue, Jo} and set B = {Bob, Tom, Moe}, find BxA.
12. Use Venn diagrams to deteraiine whether ^ U (5 fl C)' =
U 5') U
U C ) for all sets A, B, and C.
Show your work.
13. A recent survey of 510 people revealed the following information.
175 people like the color green
227 people like the color red
285 people like the color blue
100 people like the colors green and red
96 people like the colors green and blue
87 people like the colors red and blue
59 people like all 3 colors
( 3S
Construct a Venn diagram and then determine
a) the number of people who only like the color green.
b) the number of people who only like 1 color.
c) the number of people who like the colors red and blue, but not green.
2^
d) the number of people who like the colors green OT blue, but not red.
^jb^
e) the number of people who only like exactly 2 colors. \ O C c
14. Show that the set has a cardinality of N \
H
-
^
15. Determine whether each number is rational (R) or irrational (I).
a) 0.532
_ t
b) A
^
c) 0.646879347687937
d) v ?
e) V36
?-
16. State the property that each statement illustrates.
a) a{2^b)^{a.2)Ha-b)
fltMU^VL
b) a + {2 + h) = {a + 2) + b
iVx^Xjd^Nl^-
17. Without using a truth table, determine the truth value of the given compound statement.
Easter is in December and Christmas is in May, or Christmas is not in May and Halloween is in
A in iviuy, ur K^rirmiinux la nui in iviuy uri
October.
p: Easter is in December (F)
q: Christmas is in May (F)
r: Halloween is in October (T)
18. Construct a true table for the following statement.
Your dinner includes soup, or salad, or a baked potato and salad.
9
r
T
T
T
T
T
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
T
F
F
F
T
T
?
T \\
r
T
T
T
f
T n \ T
T
r
V\
f
7.
19. Write an equivalent statement using De Morgan's Laws, the fact that (p -> q) o {- pvq),
or the fact
that ~ (/? -> ^) o (pA ~ q).
It is not true that both 2^3 and 1 + 1 = 2.
p: 2<3
q: 1 + 1 - 2
^<3 er
20. Write an equivalent statement using De Morgan's Laws, the fact that (/? -> ^) <=> (~ p v ^ ) , or the fact
that ~ (p ^ g') <=> (pA ~ q).
It is not true that if you don't eat your meat, then you will have some pudding.
p: you eat your meat
q: you will have some pudding
uai otet^
g^ux x^^fk aoci oat
21. Write the converse, inverse and contrapositive of the following statement.
If you do, then I do.
p:
ua( do
Xf X c f c ^ - H x H CjO-C
(^^p
a
Converse:
b
Inverse: - v p ^ ^ ^ f ^
j j
c. Contrapositive: r>^C^-^^p
tJ^^^t^'f/H^/l
^
H
^ O n V .
-jr ( l ^ a ' t / " H ^ l f ^ H ^ O l ^
22. Determine whether the two statements, {- p\f -q)-^r
and ~(pAq)^r
are equivalent using truth
tables.
P
9
r
T
T
T
T
T
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
T
F
F
F
^ FV T
T
r T T T
TT F T
TT r T
T T r T
T T
r
T
F
T
f
T
F
F
f
T
rrT\
r
f
T
T
r
V-
f
^
T f
2-
A
23. Write an argument in symbolic form. Then use a truth table to determine whether the argument is valid
or invalid.
If the water is boiling, then the pan is ready. The water is not boiling and the pan is ready. Therefore,
the pan is ready.
p:
omf %
bc^^
1
T
T
F
F
T
F
T
F
T
T
I
24. Determine whether an argument is valid or invalid by comparing its symbolic form with one of the
standard symbolic forms. State the name of the standard form you are using in the comparison.
F a l l a c y of t h e C o n v e r s e
F a l l a c y of t h e I n v e r s e
P ~> Q
L a w of D e t a c h m e n t
P Q
~ P
5
p^q
p
••• P
L a w of C o n t r a p o s i t i o n
P Q
~(?
• q
L a w of S y l l o g i s m
P->Q
••• P
25. Determine whether two circuits are equivalent.
_9
_9
9
r
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
T
T
r
T
pvq
~ P
Q -> r
If I'm hungry, then I will eat. I ate. Therefore, I'm hungry.
P
Disjunctive Syllogism
26. The local club charges a $12.50 cover charge to get in, and then $5.50 per drink after that. How many
drinks can you aftbrd on a $51 budget?
Let X - \\ii)htrejr(\f^t.h
v..
27. Joe Gambler lost $2 more than three times as much money on Saturday than he did on Friday.
All together, he lost $506. How much money did he lose each day?
Equation:
28. A rectangular garden measures 3 yards by 5 yards? A gardener wants to place a border of bark of width
X around the garden. He has enough bark to cover 20 square yards. How wide of a border should be
made so that the border uses all the bark and what are the dimensions of the garden and border?
A f
Let
X =
home
3 yards
5 yards
'
Equation:
-20
• ^
4
^
V
29. The function for the height of a ball thrown into the air is h(t) = -\6t^+ 4St + 160, where hit) is the
height of the ball after t seconds.
a. Estimate the height of the ball after 2 seconds.
b. Estimate the time it takes for the ball to be 132 feet above the ground. Round^the nearest tenth of a second.
^-C'l^)
-^2,
"32- *
30. The exponential graph below shows the number of world-wide shipments of smart handheld devices,
P(t), in millions, for t years between 1999 through 2004, so 0 < ^ < 5. The general formula P(t) = P^e'^
models this growth,
a. Use the graph to find the initial population, P^.
Worldwide Sliipmenis of Smart
Handheld Devices
43.54
b. Use a point on the graph and the general formula
P(t) - P/^\Q find the constant grow rate, k .
Round lo the third decimal place.
30.23
L/
20.98
\1
'J
C
7.02
5
"
c. Use /*o and k to find the general formula/"(Q = Pfi .
""^
Soiiiir:
(
14.57
10.11
•99 '00
'01
•02
03
'04
Inlum;i!)>)iKi! Djiui Corp.: M S N \l«incy( cnlr:il;
( S I i n c . .tdiiilioiuit rc'.c.Hk'll
d. Use the formula found in part c. to predict the number of shipments in 2005. Round to nearest tenths
e. Use the formula found in part c. to find when the number of shipments will reach 1163 million
shipments. Round to the nearest year.
Q ^C'StT
31. The radioactive element, Cobalt-60, decays at an exponential rate. If there are originally 800 grams of
Cobalt-60, then the amount of Cobalt-60, P(t), remaining after t years is modeled by the formula
PCO^SOCfe", where/t<0, since the amount Cobalt-60 decreases as time goes on.
a. If approximately 400 grams of Cobalt-60 remain after 5.3 years, find the decay rate, k. And, state the
function that models this case. Round to the third decimal place.
5 ^
b. How much Cobalt-60 will remain after 11 years? Round to the nearest gram.
c. How long will it take to have only 1 gram remain? Round to the nearest year
gEC
" ^
^
32. If P(A)=0.2, P(B)=0.3, and P(A or B)=0.45, then find P(A and B).
- c C ^
-
-
X
^
33. Two fair dice are rolled. Find the probability of each of the following:
a. P(the sum of the two numbers is 9). g ^
^
b. P(the sum of the two numbers is odd).^^^
\3!r
c. P(the sum of the two numbers is greater than 9)__.—
d. P(the sum of the two numbers is odd A N D greater than 9). p^^^xVlC)" VT^^^ / cdS>i
e. P(the sum of the two numbers
)ers is odd OR greater than 9).
3fc
'
v
\<. A-^cK
34. ONE card is drawn at random from a standard 52-card deck. Find the probability of each of the
following:
^—r—
a. P(the card is a a).
- \ - A -
b. P(the card is not a *).
^
•
c. P(the card is a number greater than 5 and less than 9).
d. P(the card is a number greater than 5 and less than 9 OR
52
5Z
5Z
.
5Z
e. P(the card is a number greater than 5 and less than 9 AND ¥ ) .
heater, or baseboards. The sample space is listed below.
a. Find P(2 bathrooms). Qj
S;iinji!c .Space
35. Houses are listed as 1, 2, or 3 bedroom homes, with 1 or 2 bathrooms, and have either a fireplace, a
b. Find P(not having 2 bathrooms or baseboards).
I B '
c. Find P(2 bathrooms / there are 2 or 3 bedrooms).
XL
IzA
d. Find P(fireplace / there are 2 bedrooms).
36. A bag contains 7 cherry, 5 orange, and 3 lemon candies.
a. You reach in and take 3 pieces of candy at random at the same time. Find P(2 cherry and 1 lemon).
,563
^55 ^?5/65
b. You reach in and take 3 pieces of candy at random one at a time without replacing the any of the
candies. Find P(r' cherry, 2°*^ cherry, 3''^ lemon).
37. A license plate is to consist of 3 letters followed by 2 digits. Determine the number of different license
plates possible if:
a. The 1^' letter has to be a vowel {a, e, i, o, u) and the 1^' number has to be greater than 6. Also,
repetition of letters and numbers is permitted.
The 1^' letter has to be a vowel (a, e, i, o, u) and the 1^' number has to be greater than 6. Also,
repetition of letters and numbers is NOT permitted.
38. You want to arrange 6 books (math, science, fiction, history, biology, and English) on a bookshelf
a. How many different ways can all 6 books be arranged?
b. How many different ways can the 6 books be arranged i f science is first, math is in the middle and
history is on the end?
c. How many different ways can the books be arranged if you threw out 2 books and only kept 4 of the
6 books?
•
\
^
^
V_
6j^^\
39. Say there were 8 women and 9 men applying for 7 positions at a company.
a. Find the number of different combinations the company can hire the 7 people.
b. Find the number of different combinations the company can hire the 7 people, if 4 have to be women
and 3 have to be men.
40. Below is a list of the current gas prices in Santa Cruz from GasPriceWatch.com
1
Price
Price
Station
a. Create a frequency distribution for the data.
Arco
705 Capitola Rd
Ave
Santa Cruz, C A 95062
$3.59
Costco (Membership Required)
220 Sylvania Ave Harvey W Blvd
Santa Cruz, C A 95060
$3.59
Valero
1319 Ocean St eFranidin St
Santa Cruz, C A 95060
$3.63
SavOn
1415 Ocean St © P r y c e S t
Santa Cruz, C A 95060
$3.63
e7th
Frequency
7
/
5
3
4
.... .
5
L
5 f-b
6
.
$3.63
Valero
1701 Capitola Rd
Santa Cruz, C A 95062
$3.63
ei7th Ave
\
7
Kwik Serv
1725 Mission St (SR-1) ©Palm SI
Santa Cruz, C A 95060
S
b. Create a histogram for the data.
10
11
13
14
15
16
17
$3.65
Valero
2202 Mission St (CA-1) ©Miramar DR
Santa Cnjz, C A 95060
$3.65
Rod & Ros
1024 Water St Seabright Ave
Santa Cruz, C A 95062
$3.65
US Pro
3,51
5.65
hvf\
$3.65
Live Oak American
1200 17th Ave e B r o m m e r St
Santa Cruz, C A 95062
$3.65
76
904 Mission St ©Walnut Ave
Santa Cruz, C A 95060
$3.75
Rotten Robbie (#35)
1906 Mission St ©Berkshire Ave
Santa Cruz, C A 95060
$3.75
76 - Steve's
1500 Soquel Dr ©Paul Svreet Rd
Santa Cruz, C A 95065
$3.75
76
805 Ocean St ©Water St
Santa Cruz, C A 95060
$3.79
Shell-Gas & Car Wash
2700 Soquel Ave
Ave
Santa Cruz, C A 95062
$3.79
Chevron
404 Soquel Ave Ocean St
Santa Cruz, C A 95062
$3.83
e7th
18
$3.63
Valero
335 Mission St (CA-1) gHighland
Santa Cruz, C A 95060
823 Mission St ©Walnut Ave
Santa Cruz, C A 95060
12
_..
Rod & Ros Gas Mart
1024 Water St eSeabright Ave
Santa Cniz, C A 95062
c. Create a circle graph for the data.
d. Find
' ind the
the mean
mean of
ol the
me data.
uaia.
/
\>
n i \i C^o
6.
e. Find the median of the data.
Zl
f
2-
Find the mode of the data.
g. Find the range of the data
h. Find the standard deviation of the data.
Data
Data-Mean
(Data-Mean)^
3 ^
Veer 9
M
AVI
^Oli^l
it)
3.^
Total:
Standard Deviation:
s=
i.
n-l
Pretend that the distribution is normal. Using the z-score, find the number of gas stations that lie
between the first standard deviation on the left and the third standard deviation on the right.
^ ^ ^ ^ ^
,VfS^
1^B> - ^^^^^^
^'i-^^-^