3
3 Math Test Test – No Calculator Calcu lator 25 MINUTES, 20 QUESTIONS Turn to Section 3 of your answer sheet to answer the questions in this section.
DIRECTIONS For questions 1-15 1-15,, solve each problem, choose the best answer from the choices provided, and �ll in the corresponding circle on your answer sheet. For questions 16-20 16-20,, solve the problem and enter your answer in the grid on the answer sheet. Please refer to the directions before question 16 on how to enter your answers in the grid. You You may use any available space in your test booklet for scratch work.
NOTES 1. The use of a calculator calculator is not permitted. permitted. 2. All variables and expressions used represent represent real numbers unless otherwise indicated. 3. Figures provided provided in this test are drawn to scale unless otherwise indicated. 4. All �gures lie in a plane unless otherwise indicated. 5. Unless otherwise indicated, the domain of a given function f is is the set of all real numbers x for for which f ( x ) is a real number. REFERENCE
•
ℓ
r
h
w A = ℓw
A = π r 2 C = 2 π r
a 2 2 2 c =a +b
b A = 1 bh
2x
c
b
30
• r
r
°
x
s
45
°
s
°
45
h
h
V = ℓwh
V=
πr
2
h
4
= 3 πr 3 V =
ℓ
1 V = = 3 π r 2h
The number of degrees degrees of arc in a circle is 360. The number of radians radians of arc in a circle is 2 . The sum of the measures measures in degrees degrees of the angles of a triangle is 180.
36
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Cracking the New SAT
w
r
ℓ
°
3 Special Right Triangles
h w
2
s
x
2
h
60
1 = 3 ℓwh V =
3 1
3 3 If 3x + y = 4, and y = 5x – 8, then 8 x equals
What is one possible solution to the equation 36 6 =3 ? − m+2 m−2
A) 0 B) 1
A) 2
4 3 D) 12 C)
B) 3 C) 4 D) 5
4 2 If −3x + 2 < 11, then which of the fol lowing accurately expresses all possible values of x ?
If z 2 – 4z – 45 = 0, then which of the following sets includes all possible values of z ? A) {9}
A) x < −9
B) {–5, –9}
B) x > −3
C) {–5, 9}
C) x < −3
D) {–9, 5}
D) x < 3
3
3
5
7 2w + 2s = 14 3w – 2s = 1 In the system of equations shown above, what is the value of w ? A)
1
B)
3
C)
4
Which of the following could be the graph of the equation 3x – 6 y = 4 on the xy -plane? A)
y
4, 0 3
x
D) 15
0,
B)
− 2 3
y
6 A chemistry student places 10 viable bacteria in a petri dish. Every hour a fer the initial placement, she counts the number of bacteria in the dish. She notices that the number of bacteria, n, doubles with each passing hour, h. Which of the following functions could be the model for the number of bacteria in the dish over time?
(0, 4)
x
− 4 , 0 3
A) n(h) = 10h + 10 B) n(h) = 10(h + 1)2 C) n(h) = (10h)2
C)
y
D) n(h) = 10 × 2h 0, 2 3
D)
4, 0 3
x
y
(0, 4) 4, 0 3
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Cracking the New SAT
x
3
3
8
10 Rachel walks from home to school every morning. Her distance from school, D, in meters, afer t seconds of walking is represented by the function D(t ) = 475 – 2.5t , where 0 ≤ t ≤ 190. In this function, what does the constant 475 represent?
48
A) Rachel’s walking speed, in meters per second
24
B)
10
Te amount of time it takes Rachel to walk from
home to school 12
C)
Te �gure
above shows a water trough with two trapezoidal ends and a rectangular base with a length of 48 inches and a width of 12 inches. Te top edges of the trapezoidal ends are 24 inches long, and the slanted sides are 10 inches long. What is the volume of water that can be contained in this trough if it is �lled to the top, in cubic i nches? A)
6,912
B)
8,640
Te
distance from Rachel’s house to school
D) How many steps Rachel takes when walking to school
11
y
C) 11,520 D) 13,824
5
c(x) a(x) d(x) 0
–5
5
x
b(x)
9
–5
A company’s pro�t made from selling an item will depend on the price of the item, x , in dollars. Te function P (x ) = –2x 2 + 20x – 48 can be used to determine the company’s pro �t, in thousands of dollars. Which of the following is the company’s maximum pro�t for selling this item? A) $1,500
A system of four equations and their graphs in the xy plane are shown above. If e is a mathematical constant equal to approximately 2.718, which of the following could be the equation of the function d (x ) ?
B) $2,000
A) d (x ) = –3 + ex
C) $3,000 D) $4,800
B) d (x ) = 3 – e–x C) d (x ) = D) d (x ) =
2 1 + e −2 x 2 1 e2 x
3
3
12
13
MOTOR
ROPE WINCH
BELT
Te �gure
above shows a simple pulley system in
which a motor with a small pulley drives a belt to
A) 5
turn a pulley on a rope winch, lowering the rope. r , where r is Te velocity ratio, V , is de� ned as V = m the diameter of the rope winch pulley and m is the
B) 10
diameter of the motor pulley. Te output speed, O, of the system is de�ned as O = i , where i is the input V speed of the system. Which of the following gives the value of m in terms of i, O, and r ?
Or A) m = i B) m =
ir O
Oi C) m = r D) m =
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A student is studying how his classmates spend their time each day. He asks each classmate to separate an average day into six categories, noting how many hours per day are spent on each activity. Te student has broken the activities into categories of School, Work, Meals, Homework, Sleep, and Entertainment. Te student then averages the responses and compiles the results. He discovers that, in a typical day, his classmates break down their days in the following way: 25% School, 17% Work, 8% Meals, x % Homework, 25% Sleep, and the rest on Entertainment. He represents this data in a pie graph with radius 10 centimeters. If the central angle of the Homework sector is 36˚, how much greater is the area of the Entertainment sector than the area of the Homework sector?
i Or
Cracking the New SAT
C) 15 D) 25
3
3
14
15
y = –x + a
Te number of gems a player has ear ned in a match
of Diamonds on the Aegis is modeled by the function m g ( c , l , m,o ) = c ( 200 + 9l ) + − 300o + 625, where c 100 is the number of Combos made, l is the average level of each Combo, m is the number of minutes since the beginning of the match, and o is the number of
y = –x 2 + b x 2 + y 2 = 13 In the system of equations above, a and b are positive constants. Te system of equations has two solutions. Which of the following could be the graphs of these equations in the xy -plane? A)
y
Oopsies committed by the player. What does the 5
number 100 represent in this function? A)
Te number of gems the
player begins a match
with. B)
Te number of minutes since the
–5
D)
5
x
beginning of the
match. C)
0
–5
Te number of gems earned per
minute, excluding gems from Combos and starting gems.
B)
y
Te number of minutes per gem
earned, excluding gems from Combos and starting gems.
5
–5
0
5
x
–5
y
C) 5
–5
0
5
x
–5
D)
y 5
–5
0
–5
5
x
3
3
19
20 Te function
y
q is de�ned by q(x ) = 4x 3 – bx 2 +14x + 12,
where b is a constant. In the xy -plane, the graph of q 1 2
contains the points (– , 0), (2, 0), and (n, 0). What is the value of b ? x
(–18, –5)
(–8, –4)
Note: Figure not drawn to scale. In the xy -plane above, what is the x -intercept of the given graph?
ST O P If you finish before time is called, you may check your work on this section only. Do not turn to any other section in the test. 44
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Cracking the New SAT
4
4 Math Test – Calculator 25 MINUTES, 20 QUESTIONS
DIRECTIONS
For questions 1-30, solve each problem, choose the best answer from the choices provided, and �ll in the corresponding circle on your answer sheet. For questions 31-38, solve the problem and enter your answer in the grid on the answer sheet. Please refer to the directions before question 31 on how to enter your answers in the grid. You may use any available space in your test booklet for scratch work.
NOTES
1. The use of a calculator is permitted. 2. All variables and expressions used represent real numbers unless otherwise indicated. 3. Figures provided in this test are drawn to scale unless otherwise indicated. 4. All �gures lie in a plane unless otherwise indicated. 5. Unless otherwise indicated, the domain of a given function f is the set of all real numbers x for which f ( x ) is a real number. REFERENCE
•
ℓ
r
h
w A = ℓw
A = π r 2 C = 2 π r
a 2 2 2 c =a +b
b A = 1 bh 2 • r
h
2x
c
b
r
30
60
°
x
s
45
°
s
°
45
°
s
x
3 Special Right Triangles
h
h
h
ℓ V = ℓwh
V=
πr
2
h
4
V = 3 πr 3
ℓ
1 V = 3 π r 2h
The number of degrees of arc in a circle is 360. The number of radians of arc in a circle is 2 . The sum of the measures in degrees of the angles of a triangle is 180.
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Cracking the New SAT
w
r
w
2
1 V = 3 ℓwh
4 1
4 3
Each student at a high school throws away two pounds of garbage a day. If there are s students at the school and the non-students at the s chool throw away a total of 350 pounds of garbage a day, which of the following expressions represents how many pounds of garbage are thrown away at the school each day?
If w > 3, then which of the following must be true? A)
−w < −3
B)
−w > −3
C) w = 4 D) w is an integer
A) 2(s + 175) B) 2(s + 350) C) 350(s + 2) D) 350s 4 A roller coaster at a carnival has 18 cars that hold two people each. Te ratio of working seatbelts to broken seatbelts is 8 to 1. If no one can sit in a seat with a broken seatbelt, what is the maximum number of people that can ride t he roller coaster at once?
2 Rayyan wants to make sure that he is at his target heart rate when he exercises. His ideal heart rate for heart health is 55% of his maximum heart rate, de�ned as the di ff erence between the number 220 and his current age. If Rayyan is currently 20 years old, what is his target heart rate? A) 101 B) 110 C) 121 D) 132
A)
4
B) 16 C) 32 D) 36
4
4
5
6 Growth of a Bacteria Colony
3x − 2 4 A) −2 If
80 70
a i r e t c a B e l b a i V f o r e b m u N
=
x , then what is the value of x ?
60 50
B)
0
C)
1 3
D)
2
40 30 20 10 0
1
2
3
4
Time (hours)
Te
graph above shows the growth of a bacteria colony in a petri dish over time. Which of the following is true about the graph? A)
Te number of viable bacteria
in the colony
doubles every hour. B)
Te number of viable bacteria
in the colony increases most rapidly during the �rst hour.
C)
Te number of viable bacteria
D)
Te number of viable bacteria in the colony will
in the colony increases at a constant rate of 5 per hour. be 100 at t = 5.
7 Jeff erson is conducting an experiment to determine how the height of water in a beaker is a ff ected if he adds marbles to the water. He starts with an empty beaker and adds some volume of water. He then adds marbles one at a time. Te height of t he water, H , in centimeters, a fer he adds m marbles is represented by the function H(m) = 5 + 0.2m. How many marbles must Jeff erson add to double the original height of the water in the beaker? A)
5
B) 10 C) 20 D) 25
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Cracking the New SAT
4
4
8
10 One lit match releases approximately 1,100 joules of heat energy. If 1 joule equals 0.239 calories, how many matches must be lit to produce 2,100 calories of heat energy? A)
2
B)
3
C)
8
y
x
D) 62
9 If 1 < r < 4 and 0 < s < 5, then what is the range of r + s ? A)
−4
< r + s < 1
B)
0 < r + s < 5
C)
1 < r + s < 9
D)
0 < r + s < 10
Which of the following could be the equation of the graph in the xy -plane shown above? A) y = 2x B) y = −2x – 2 C) y = −2 D) y = 2x – 2
4
4
11
12 A student is graphing the potential energy, in kilojoules, of a certain chemical reaction. At the start of the reaction, the reactants have a constant level of potential energy. As the reactants approach transition state, the potential energy increases, then drops o ff as the reaction nears completion. Te resulting products have less potential energy than the original reactants did. Which of the following could be the graph of the potential energy of this chemical reaction? A) ) J k ( y g r e n E l a i t n e t o P
Reaction Pathway
B) ) J k ( y g r e n E l a i t n e t o P
Reaction Pathway
Birth records were examined for all babies born in the United States in the year 2006. Social scientists analyzed the number of babies born in each region of the country based on birth month. Te table below shows the data for all four regions for a portion of the year. Birth Month by Region (in thousands) July
August
September
Total
Paci�c
114
97
74
285
Mountain
103
88
69
260
Central
49
120
180
349
Eastern
121
115
70
306
Total
387
420
393
According to the table, if a baby was born in the United States during one of the three months shown, what is the probability that the baby was born in August in the Central region? A)
1 10
B)
3 20
C)
1 5
D)
3 7
C) ) J k ( y g r e n E l a i t n e t o P
Reaction Pathway
D) ) J k ( y g r e n E l a i t n e t o P
Reaction Pathway
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Cracking the New SAT
4
4
13
15 If −1 ≤ p ≤ 3 and −4 ≤ q ≤ 3, then what is the range of p – q ? A)
−5 ≤ p –
q ≤ 0
B)
−4 ≤ p –
q ≤ 3
C)
−4 ≤ p –
q ≤ 7
D)
−5 ≤ p –
q ≤ 6
x What is the value of if 3x + y = 16 and y 3x + 2 y = 23 ? A)
3 7
B)
16 23
C)
7 3
D) 21 14 Social science researchers want to conduct a survey about modes of interpersonal communication in the digital age. Tey want to survey a random sample of people and apply the results to the general population.
16
To ensure that their results are reasonably accurate, they want a 95% con�dence level, meaning that the survey results only have a 5% chance of di ff ering from 1 the true population average. Te expression , N where N is the number of participants in the survey sample, is an estimation of the margin of error at a 95% con �dence level. Which of the following is the closest to the minimum value of N needed for this survey? A)
5
B)
20
C) 400 D) 950
If 4 x + 21 = x , then which of the following represents all possible values of x ? A) {–3} B) {7} C) {–7, 3} D) {–3, 7}
4
4
17
19 Mike’s Meatball Market sold 2,321 meatballs from Friday to Sunday. All the meatballs sold at Mike’s are either beef or turkey. Beef meatballs sell for $1.50 each, turkey meatballs sell for $2 each, and the total sales from beef and turkey meatballs was $3,876. If the number of beef meatballs sold is represented by b, and the number of turkey meatballs sold is represented by t , then which of the following systems of equations, if solved, would determine the number of each type of meatball sold?
Line l passes t hrough the points (0, 3) and (2, 6). Line m has the same y -intercept of line l but has a slope that is half the slope of line l . Which of the following points lies on line m ? A) (2, 9) B) (4, 6) C) (2, 6) D) (4, 15)
A) 1.5b + 2t = 3,876 b + t = 2,321 B) 1.5b + 2t = 2,321 b + t = 3,876 C)
b t + = 3, 876 1.5 2 bt = 2,321 3, 876 3 b + t = 2,321
D) 1.5b + 2t =
20 At a certain company, 25% of the employees hold master’s degrees. Of t hose employees who do not hold master’s degrees, 10% are employed in entry-level positions. No one with a master’s degree is employed in an entry-level position. Which of the following could be the number of employees holding entry-level positions in the company? A) 5 B) 6 C) 7 D) 8
18 A certain 3D printer creates objects by building layers on top of each other. Te average layer thickness is 102 micrometers. Tere are 1,000,000 micrometers in a meter, and 1 inch equals 0.0254 meters. Approximately how many layers are needed to print an object one inch thick? A)
250
B)
630
C) 2,540 D) 4,015
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Cracking the New SAT
