Download 2012 State Math Contest Wake Technical Community College

2012 State Math Contest
Wake Technical Community College
Algebra II Test
1. The population of Tennessee was approximately one million people the beginning of 1850. If the
population trend indicates that it has tripled every 90 years since 1850, during what year would the
population first reach 12.6 million?
a. 2057
b. 2058
c. 2059
2. Determine the domain of the function ( )
a. (
]
[
)
b. (
3. A line L is perpendicular to
intersection between the line
a. (
)
b. (
)
)
d. 2060
√
e. 2061
.
c. (
)
d. [
]
and passes through the point (
and the line L.
c. (
)
d. (
e. (
)
). Determine the point of
)
e. (
)
4. Tom, Dick, and Harry each have at least one child. The average of Dick’s and Harry’s children plus the
number of Tom’s children is 5. The average of Dick’s and Tom’s children added to the number of
Harry’s children is 7. Determine the total number of children in the three families.
a. 8
5. If
b. 9
is a factor
a. 18
b. 22
c. 10
, determine
a. Empty Set
b. A point
)
c. Two intersecting lines
1
e. 12
d. 16.5
e. 15.5
.
c. 15
6. Which of the following best describes the graph of (
d. 11
?
d. Two parallel lines
e. A circle
2012 State Math Contest
Wake Technical Community College
Algebra II Test
7. A basketball player is successful on 40% of the first 25 free throws she shoots in a season. Determine
the minimum number of consecutive free throws she must be successful on to raise her percentage to
at least 67%.
a. 14
b. 15
c. 20
d. 18
e. 21
8. A bridge charges 2-axled vehicles a $5 toll and 3-axled vehicles an $8 toll. In an hour the bridge
collected $741 from 120 vehicles. If tolls were $1 higher for 2-axled vehicles and $2 higher for 3-axled
vehicles, how much would the bridge have collected?
a. $888
b. $908
c. $926
. Determine (
9. Assume the operation © is defined by
a.
b.
10. Let ( )
d. $934
c.
. Determine (
a.
) (
(
d.
e. $1012
)) for all
.
e.
).
b.
c.
d.
e.
11. A population consists of the 3920 employees at Rex Hospital. A random sample of 2.5% of the
employees was asked if they would take a statistics course if one were offered on site. Eight of the
employees sampled said “yes.” About how many of the 3920 employees would likely take a statistics
course, based on the sample data?
a. 328
b. 325
c. 320
2
d. 345
e. 340
2012 State Math Contest
Wake Technical Community College
Algebra II Test
12. A metal rod is heated to 400°F and then placed in a cooler set at −10°F, where the rod’s temperature
drops to 300°F after the first 2 minutes. Assume the rod cools at an exponential rate. Determine the
average rate of temperature drop during the first 5 minutes in degrees F per minute.
a. −41.24 °F/min
b. −41.03 °F/min
c. −41.09 °F/min
d. −41.15 °F/min
e. −41.35 °F/min
13. The cumulative box office revenue from the movie Finding Nemo can be modeled by the logarithmic
(
)
function ( )
where is the number of weeks since the movie opened
and ( ) is given in millions of dollars. According to the model, how many weeks after the opening of
the movie was the cumulative revenue equal to $300 million?
a. 9.7 weeks
b. 8.8 weeks
c. 8.9 weeks
d. 10.1 weeks
e. 9.8 weeks
14. Which of the following is equivalent to
a.
b.
c.
d.
e.
15. In a baseball league there are twelve teams. If each team plays each other team twice, how many
games must be scheduled?
a. 66
b. 132
16. The solution set of
a.
c. 144
d. 72
e. 121
is a subset of which if the following inequalities?
b.
c.
d. b and c
3
e. a, b, and c
2012 State Math Contest
Wake Technical Community College
17. Suppose ( )
( )
and ( (
a. 5
and ( )
))
( ( ))
b. 12
Algebra II Test
for integers
and . What is the product of
and
if
?
c. 48
d. 182
e. 210
18. Mitch traveled one hour longer and two miles farther than Calvin, but averaged 3 mph slower. If the
sum of their times was four hours, what was the sum of the distances they traveled?
a. 26 miles
b. 5 miles
c. 28.5 miles
d. 46 miles
e. 30.5 miles
19. A basketball team scores 78 points on 41 baskets (field goals worth 2 points, free throws worth 1
point, and 3-point shots). If the number of each type of basket is different, and the number of baskets
of any two types differs by no more than 4, how many field goals are scored?
a. 11
b. 12
20. Let
a.
c. 13
and
b.
d. 14
, determine
c.
e. 15
.
d.
e.
21. In a group of 30 students, 25 are taking math, 22 are taking English, and 19 are taking history. If the
largest number who are taking all three is and the smallest number who could be taking all three is
, determine the sum of and .
a. 24
b. 25
c. 26
d. 27
4
e. 28
2012 State Math Contest
Wake Technical Community College
Algebra II Test
22. If , , , and are nonzero numbers such that and
are solutions to
, determine
a. −2
b. −1
are solutions to
.
c. 0
d. 1
and
and
e. 2
23. In a recent soccer competition, each of three teams played the other teams once. Statistics for the
three games are displayed in the table. What was the score of South versus East game (give South’s
goals first)?
Team
South
North
East
a. 2 - 0
Goals Scored
By Team
6
3
2
Wins Losses Ties
1
1
0
b. 2 - 1
0
0
2
1
1
0
c. 3 - 1
Goals Scored
Against Team
4
2
5
d. 3 - 2
e. 4 - 2
) of prime numbers for which the equation
24. Let S be the set of all ordered triples (
has at least one rational solution. How many primes appear in S at least seven
times?
a. An infinite number
b. 0
c. 1
d. 2
e. 3
25. How many different 3-letter strings can be formed from the letters of TENNESSEE (no letter can be
used in a given string more times than it appears in the word)?
a. 46
b. 48
c. 52
5
d. 57
e. 50
2012 State Math Contest
Wake Technical Community College
Algebra II Test
SHORT ANSWER
Place the answer in the appropriate space.
66. Dr. Scott has 3 types of candy to pack in gift boxes: chocolate truffles, peppermint sticks, and
chocolate-covered almonds. Each box must contain exactly 8 pieces and at least one of each type of
candy. How many different gift boxes can Dr. Scott create?
67. The equation
integers. What is the value of | |
has four solutions (
| | | | | |?
) in which both coordinates are
68. Matthew knows that one of five books he was reading contains his lost concert ticket. He decides to
look for it. What is the expected (average) number of books he will need to search before he knows
which book holds the ticket?
69. Two boys have to travel 25 miles, but only have one bike between them. David walks at 2.5 mph and
Fred walks at 6 miles per hour. Both cycle at 12.5 miles per hour. Only one boy is allowed on the bike
at a time. What is the shortest time needed for them both to be at their destination? Give your
answer in hours and minutes.
70. From the series of the first one hundred odd integers, it is possible to extract sub-series which sum to
almost any four digit power. For example,
What is the first term of such a sub-series which sums to
6
?
2012 State Math Contest
Wake Technical Community College
Algebra II Test
1. a
2. d
3. e
4. b
5. d
6. c
7. e
8. b
9. a
10. d
11. c
12. a
13. e
14. d
15. b
16. a
17. b
18. e
19. c
20. d
21. b
22. a
23. e
24. c
25. c
66. 21
67. 44
68. 2.8
69. 3 hours and 42 minutes
70. 101
7
2012 State Math Contest
Wake Technical Community College
Algebra II Test
be the years since the beginning of 1850, then ( )
1. Let
Solving
gives
or during 2057.
gives the domain to be [
2.
models the population in millions.
].
solving gives the point of intersection as (
3. The equation of line L is
).
4. Let be the number of Tom’s children, let be the number of Dick’s children, and let be the number
of Harry’s children. Then
and
or
and
.
Subtracting the two equations gives
. If
, then
and
. If
, then
and
, which is impossible. Hence, the total number of children is 9.
5. Since
(
is a factor
)
.
)(
, then
. This gives
(
. Hence,
)(
and
) and
. Thus,
)
6. (
is equivalent to
. The solutions to this equation all lay either on the x-axis
or the y-axis which are two intersecting lines.
7. Let be the number of required free throws, then
throws.
8. Let
9. (
or 21 free
be the number of 2-axled vehicles and let be the number of 3-axled vehicles. Then
( )
and
. Solving gives
and
. Hence ( )
.
) (
10. (
11.
. Solving gives
(
)
(
))
(
(
)
(
)
(
) (
)
)
.
.
)
employees were sampled. Hence the sample suggests that
employees would take the statistics course.
(
)
12. The limiting value of the temperature of the rod is −10°F. Exponential functions have a limiting value
of zero so the data must be shifted vertically first before determining an exponential model. Using the
) (
)} gives the exponential model
regression on the calculator for the two points {(
( )
(
). Shifting this down again gives ( )
(
)
where is
the number of minutes and
/min.
13.
(
)
is the temperature in degrees Fahrenheit. Hence,
gives
weeks.
8
( )
( )
2012 State Math Contest
Wake Technical Community College
(
14.
)
Algebra II Test
.
(
15. If each team played the other teams once, they need to schedule
play each team twice they will need to schedule 132 games.
is (
16. The solution set of
(
]
[
). The solution set of
is (
of the solution set of
17.
]
]
and ( (
and solving gives
)
and
[
)
[
)
games. So in order to
). The solution set of
is (
)
[
is
). The solution set of
). Hence, the solution set of
.
is only a subset
( (
)
)
or
. Thus, the product is 12.
. Substituting
18. Since their total time was 4 hours Mitch traveled 2.5 hours and Calvin traveled 1.5 hours. Let be the
distance that Calvin traveled and
be the distance that Mitch traveled. Then
.
Solving gives
miles and 30.5 miles as their total distance.
19. Let be the number of free throws, be the number of field goals, and be the number of 3-point
shots. Then
and
. These two equations give
and
. Using the table feature of the calculator and the conditions given in the problem the only
). Thus, there were 13 field goals.
solution is (
20.
gives (
(
)
(
)
)
(
)(
). Since
,
.
21. It is clear that
. To determine the smallest number taking all three start with the number that
are not taking history and assume all of them are taking math. That means that 14 must be taking
both math and history. Then factor in those students who are not taking English and assume all of
them are taking both math and history. That leaves at least 6 that must be taking all three. Hence
.
22. It is easy to see that
,
and
;
;
. Hence,
; and
. Using these four equations yields
.
23. Let be the number of goals South scored against East; be the number of goals South scored
against North (which is the same as the number of goals North scored against South); be the number
of goals North scored against East; be the number of goals East scored against South; and be the
number of goals East scored against North. Then
;
;
;
;
; and
. Solving this system gives a score of 4 – 2.
9
2012 State Math Contest
Wake Technical Community College
Algebra II Test
24. Since the coefficients of the quadratic are prime numbers and there is a rational solution to the
)(
) or (
)(
).
quadratic equation, then there are only two ways to factor (
Hence
or
. Since is also prime then either
or
in every triple.
{(
) (
) (
) (
) (
)(
) (
)} is a subset of S and both 2 and 5 show up
) (
) (
) (
)}. This
seven times. Each triple has the form {(
means every prime other than 2 or 5 can show up at most six times, because for all other primes
to be in a triple one of the following must be a prime: {
} can be prime.
of {
} and only one
25. Assume all three letters chosen are different then there are 24 different strings. If the two letters are
the same there is three ways to choose the additional letter and three locations for the third letter
(eg. EET, ETE, and TEE). This adds 27 additional strings. Plus one more for EEE. That gives 52 strings.
66. This is a counting problem that amounts to determining the number of ways to add three positive
) (
) (
) (
) (
)}. If the numbers are all different
integers to get 8. The list is {(
there are six different boxes and if two of the numbers are the same there are three different boxes.
That gives 21 different boxes.
67.
factors {
{(
) (
)(
)
or (
. In order for and to be integers the
} must be equal to factors of
. Using this gives the four solutions
) (
)}. Hence | | | | | | | |
.
) (
68. Once Matthew has searched 4 books he will know which book holds the ticket. The probability that he
needs to look in exactly one book is 0.2; the probability that he needs to look in exactly two books is
0.2; the probability that he needs to look in exactly three books is 0.2; and the probability that he
needs to look in exactly four books is 0.4. That gives an expected value of
( )
( )
( )
( )
books.
69. Let be the distance that David bikes, then
is the distance that he walks. Hence, Fred will walk
miles and bike
miles. Setting the times equal gives
and
miles.
The total time is
70. Let
hours or 3 hours and 42 minutes.
be the first term. Then
(
where
gives
)
and
(
)
(
(
))
(
. This means that
.
10
(
and
))
(
)
. This