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Math Problems
Algebra 1
Write exponential functions: word problems
A huge ice glacier in the Himalayas initially covered an area of
45
45
45
square kilometers. Because of changing weather patterns, this glacier begins to melt, and the area it covers begins to decrease exponentially.
\newline
The relationship between
A
A
A
, the area of the glacier in square kilometers, and
t
t
t
, the number of years the glacier has been melting, is modeled by the following equation.
\newline
A
=
45
e
−
0.05
t
A=45 e^{-0.05 t}
A
=
45
e
−
0.05
t
\newline
How many years will it take for the area of the glacier to decrease to
15
15
15
square kilometers?
\newline
Give an exact answer expressed as a natural logarithm.
\newline
years
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The following formula gives the temperature's measure in degrees Fahrenheit
F
F
F
, where
C
C
C
is the measure in degrees Celsius:
\newline
F
=
9
5
C
+
32
F=\frac{9}{5} C+32
F
=
5
9
C
+
32
\newline
Rearrange the formula to highlight the measure in degrees Celsius.
\newline
C
=
C=
C
=
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The following formula is used in economics to find a factory's unit labor cost
U
U
U
, where
O
O
O
is the hourly output per worker and
W
W
W
is the hourly compensation per worker.
\newline
U
=
W
O
U=\frac{W}{O}
U
=
O
W
\newline
Rearrange the formula to highlight the hourly output per worker.
\newline
O
=
□
O=\square
O
=
□
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The following formula gives an object's average speed
S
S
S
, where
T
T
T
is the time it takes the object to travel a distance
D
D
D
.
\newline
S
=
D
T
S=\frac{D}{T}
S
=
T
D
\newline
Rearrange the formula to highlight time.
\newline
T
=
T=
T
=
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Kirti has developed formulas to help her determine the size of her rectangular flower garden each spring. If she wants to plant
n
n
n
varieties of flowers, she determines the length,
L
L
L
, and the width,
W
W
W
, of her garden, in feet, according to the following formulas:
\newline
L
(
n
)
=
n
+
2
L(n)=n+2
L
(
n
)
=
n
+
2
\newline
W
(
n
)
=
0.5
n
+
1
W(n)=0.5 n+1
W
(
n
)
=
0.5
n
+
1
\newline
Let
A
A
A
be the area of Kirti's flower garden if she plants
n
n
n
types of flowers.
\newline
Write a formula for
A
(
n
)
A(n)
A
(
n
)
in terms of
L
(
n
)
L(n)
L
(
n
)
and
W
(
n
)
W(n)
W
(
n
)
.
\newline
A
(
n
)
=
A(n)=
A
(
n
)
=
\newline
Write a formula for
A
(
n
)
A(n)
A
(
n
)
in terms of
n
n
n
.
\newline
A
(
n
)
=
A(n)=
A
(
n
)
=
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Every chemical element goes through natural exponential decay, which means that over time its atoms fall apart. The speed of each element's decay is described by its half-life, which is the amount of time it takes for the number of radioactive atoms of this element to be reduced by half.
\newline
The half-life of the isotope beryllium-
11
11
11
is
14
14
14
seconds. A sample of beryllium-
11
11
11
was first measured to have
800
800
800
atoms. After
t
t
t
seconds, there were only
50
50
50
atoms of this isotope remaining.
\newline
Write an equation in terms of
t
t
t
that models the situation.
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Ajay is researching how the population of his hometown has changed over time. Specifically, he learns his hometown had a population of
20
20
20
,
000
000
000
in
1990
1990
1990
, and that the population has since increased by about
8
%
8 \%
8%
every
3
3
3
years.
\newline
Ajay predicts that his town can only support a population of
50
50
50
,
000
000
000
.
\newline
Ajay is relieved to see that population has not exceeded
50
,
000
t
50,000 t
50
,
000
t
years after
1990
1990
1990
.
\newline
Write an inequality in terms of
t
t
t
that models the situation.
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Every chemical element goes through natural exponential decay, which means that over time its atoms fall apart. The speed of each element's decay is described by its half-life, which is the amount of time it takes for the number of radioactive atoms of this element to be reduced by half.
\newline
The half-life of the isotope dubnium
−
263
-263
−
263
is
29
29
29
seconds. A sample of dubnium-
263
263
263
was first measured to have
1024
1024
1024
atoms. After
t
t
t
seconds, there were only
32
32
32
atoms of this isotope remaining.
\newline
Write an equation in terms of
t
t
t
that models the situation.
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A manufacturing plant earned
$
80
\$ 80
$80
per man-hour of labor when it opened. Each year, the plant earns an additional
5
%
5 \%
5%
per man-hour.
\newline
Write a function that gives the amount
A
(
t
)
A(t)
A
(
t
)
that the plant earns per man-hour
t
t
t
years after it opens.
\newline
A
(
t
)
=
A(t)=
A
(
t
)
=
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Water is poured into a rectangular container
10
10
10
centimeters
(
c
m
)
(\mathrm{cm})
(
cm
)
long,
12
c
m
12 \mathrm{~cm}
12
cm
wide, and
9
c
m
9 \mathrm{~cm}
9
cm
high, until it is
1
20
\frac{1}{20}
20
1
full. All this water is then poured into an empty cylindrical container with a circular base of radius
3
c
m
3 \mathrm{~cm}
3
cm
. The height of the water in the cylindrical container is
y
π
\frac{y}{\pi}
π
y
centimeters, where
y
y
y
is a constant. What is the value of
y
y
y
?
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36
S
+
64
U
≥
288
36 S+64 U \geq 288
36
S
+
64
U
≥
288
\newline
Choo Kheng must travel at least
288
288
288
kilometers in a submarine in order to reach her destination. The submarine travels at a constant speed on the water's surface, and travels at a constant speed underwater. In the given inequality,
S
S
S
represents the number of hours the submarine can travel on the water's surface and
U
U
U
represents the number of hours it can travel underwater in order to reach
C
h
o
o
\mathrm{Choo}
Choo
Kheng's destination. If the submarine travels for
2
2
3
2 \frac{2}{3}
2
3
2
hours on the water's surface, what is the least number of hours the submarine must travel underwater in order for Choo Kheng to reach her destination?
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The zebra mussel, Dreissena polymorpha, filters particulate organic carbon
(
P
O
C
)
(P O C)
(
POC
)
from water as part of its feeding pattern. The concentration,
c
c
c
, of
P
O
C
P O C
POC
(in milligrams per liter) remaining in a particular bay
m
m
m
months after the introduction of a population of zebra mussels can be estimated using the following equation.
\newline
c
=
0.4
⋅
0.
9
m
c=0.4 \cdot 0.9^{m}
c
=
0.4
⋅
0.
9
m
\newline
According to this estimate, how many milligrams per liter of
P
O
C
P O C
POC
were in the bay when the zebra mussels were first introduced?
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A steel ball is traveling through water with a speed of
x
x
x
meters per second, where
x
x
x
is positive. The drag force,
F
F
F
, in newtons
(
N
)
(\mathrm{N})
(
N
)
is:
\newline
F
=
0.5
+
0.004
(
x
+
50
)
(
x
−
2
F=0.5+0.004(x+50)(x-2
F
=
0.5
+
0.004
(
x
+
50
)
(
x
−
2
\newline
At what speed in meters per second does the ball have a force of
0.5
N
0.5 \mathrm{~N}
0.5
N
on it?
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A manufacturer of chemical glassware needs to purchase a certain amount of raw material. The profit,
p
p
p
, in dollars expected from purchasing
t
t
t
tons of raw material, where
t
t
t
is positive, is:
\newline
p
=
50
,
000
(
2
t
−
1
)
(
t
−
5
)
p=50,000(2 t-1)(t-5)
p
=
50
,
000
(
2
t
−
1
)
(
t
−
5
)
\newline
What is the smallest amount of raw material in tons that the manufacturer can purchase to break even with a profit of
0
0
0
dollars?
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d
=
3.2
(
t
+
1
)
(
2
t
−
3
)
d=3.2(t+1)(2 t-3)
d
=
3.2
(
t
+
1
)
(
2
t
−
3
)
\newline
An air pump increases the oxygen levels in an aquarium and reduces the build-up of waste materials. The equation shown gives the depth,
d
d
d
, in inches of an air bubble beneath the surface of the water
t
t
t
seconds after it emerges from the air pump. After how many seconds does the air bubble reach the surface?
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Shryia read a book cover to cover in a single session, at a constant rate. After reading for
1
1
1
.
5
5
5
hours, she had
402
402
402
out of the total
480
480
480
pages left to read.
\newline
Let
y
y
y
represent the number of pages left to read after
x
x
x
hours.
\newline
Complete the equation for the relationship between the number of pages left and number of hours.
\newline
y
=
y=
y
=
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Kayden is a stunt driver. One time, during a gig where she escaped from a building about to explode(!), she drove at a constant speed to get to the safe zone that was
160
160
160
meters away. After
3
3
3
seconds of driving, she was
85
85
85
meters away from the safe zone.
\newline
Let
y
y
y
represent the distance (in meters) from the safe zone after
x
x
x
seconds.
\newline
Complete the equation for the relationship between the distance and number of seconds.
\newline
y
=
□
y=\square
y
=
□
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Noa drove from the Dead Sea up to Jerusalem. When she arrived in Jerusalem after
1
1
1
.
5
5
5
hours of driving, her altitude was
710
710
710
meters above sea level. Her altitude increased at a constant rate of
740
740
740
meters per hour.
\newline
Let
y
y
y
represent Noa's altitude (in meters) relative to sea level after
x
x
x
hours.
\newline
Complete the equation for the relationship between the altitude and number of hours.
\newline
y
=
□
y=\square
y
=
□
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Kayden is a stunt driver. One time, during a gig where she escaped from a building about to explode(!), she drove at a constant speed to get to the safe zone that was
160
160
160
meters away. After
3
3
3
seconds of driving, she was
85
85
85
meters away from the safe zone.
\newline
Let
y
y
y
represent the distance (in meters) from the safe zone after
x
x
x
seconds.
\newline
Complete the equation for the relationship between the distance and number of seconds.
\newline
y
=
□
y=\square
y
=
□
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A young sumo wrestler decided to go on a special high-protein diet to gain weight rapidly. After
11
11
11
months, he weighed
140
140
140
kilograms. He gained weight at a rate of
5
5
5
.
5
5
5
kilograms per month.
\newline
Let
y
y
y
represent the sumo wrestler's weight (in kilograms) after
x
x
x
months.
\newline
Complete the equation for the relationship between the weight and number of months.
\newline
y
=
□
y=\square
y
=
□
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Noa drove from the Dead Sea up to Jerusalem. When she arrived in Jerusalem after
1
1
1
.
5
5
5
hours of driving, her altitude was
710
710
710
meters above sea level. Her altitude increased at a constant rate of
740
740
740
meters per hour.
\newline
Let
y
y
y
represent Noa's altitude (in meters) relative to sea level after
x
x
x
hours.
\newline
Complete the equation for the relationship between the altitude and number of hours.
\newline
y
=
□
y=\square
y
=
□
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A young sumo wrestler decided to go on a special high-protein diet to gain weight rapidly and at a constant rate. After
8
8
8
months, he weighed
138
138
138
kilograms. He started at
90
90
90
kilograms.
\newline
Let
y
y
y
represent the sumo wrestler's weight (in kilograms) after
x
x
x
months.
\newline
Complete the equation for the relationship between the weight and number of months.
\newline
y
=
□
y=\square
y
=
□
Get tutor help
A young sumo wrestler decided to go on a special high-protein diet to gain weight rapidly. After
11
11
11
months, he weighed
140
140
140
kilograms. He gained weight at a rate of
5
5
5
.
5
5
5
kilograms per month.
\newline
Let
y
y
y
represent the sumo wrestler's weight (in kilograms) after
x
x
x
months.
\newline
Complete the equation for the relationship between the weight and number of months.
\newline
y
=
□
y=\square
y
=
□
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A lake near the Arctic Circle is covered by a thick sheet of ice during the cold winter months. When spring arrives, the warm air gradually melts the ice, causing its thickness to decrease at a rate of
0
0
0
.
2
2
2
meters per week. After
7
7
7
weeks, the sheet is only
2
2
2
.
4
4
4
meters thick.
\newline
Let
y
y
y
represent the ice sheet's thickness (in meters) after
x
x
x
weeks.
\newline
Complete the equation for the relationship between the thickness and number of weeks.
\newline
y
=
□
y=\square
y
=
□
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A lake near the Arctic Circle is covered by a thick sheet of ice during the cold winter months. When spring arrives, the warm air gradually melts the ice, causing its thickness to decrease at a rate of
0
0
0
.
2
2
2
meters per week. After
7
7
7
weeks, the sheet is only
2
2
2
.
4
4
4
meters thick.
\newline
Let
y
y
y
represent the ice sheet's thickness (in meters) after
x
x
x
weeks.
\newline
Complete the equation for the relationship between the thickness and number of weeks.
\newline
y
=
□
y=\square
y
=
□
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A young sumo wrestler decided to go on a special high-protein diet to gain weight rapidly and at a constant rate. After
8
8
8
months, he weighed
138
138
138
kilograms. He started at
90
90
90
kilograms.
\newline
Let
y
y
y
represent the sumo wrestler's weight (in kilograms) after
x
x
x
months.
\newline
Complete the equation for the relationship between the weight and number of months.
\newline
y
=
□
y=\square
y
=
□
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A young sumo wrestler decided to go on a special diet to gain weight rapidly.
\newline
W
W
W
represents the wrestler's weight (in kilograms) after
t
t
t
months.
\newline
W
=
80
+
5.4
t
W=80+5.4 t
W
=
80
+
5.4
t
\newline
What was the wrestler's weight before his special diet?
\newline
kilograms
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Agent Hunt is transferring classified files from the CIA mainframe into his flash drive.
\newline
S
S
S
represents the size of the files on the drive (in megabytes) after
t
t
t
seconds.
\newline
S
=
5
t
+
45
S=5 t+45
S
=
5
t
+
45
\newline
What was the file size on the drive before the transfer?
\newline
megabytes
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When Quinn got home, he turned the air conditioner on.
\newline
Q
Q
Q
represents the temperature in Quinn's home (in degrees Celsius) after
t
t
t
minutes.
\newline
Q
=
42
−
0.7
t
Q=42-0.7 t
Q
=
42
−
0.7
t
\newline
What was the temperature when Quinn returned home?
\newline
degrees Celsius
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When Quinn got home, he turned the air conditioner on.
\newline
T
T
T
represents the temperature in Quinn's home (in degrees Celsius) after
t
t
t
minutes.
\newline
T
=
42
−
0.7
t
T=42-0.7 t
T
=
42
−
0.7
t
\newline
How fast did the temperature drop?
\newline
degrees Celsius per minute
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Rachel is a stunt driver, and she's escaping from a building that is about to explode!
\newline
D
D
D
represents Rachel's remaining distance (in meters) as a function of time
t
t
t
(in seconds).
\newline
D
=
−
38
t
+
220
D=-38 t+220
D
=
−
38
t
+
220
\newline
What is Rachel's speed?
\newline
meters per second
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A lake near the Arctic Circle is covered by sheet of ice during the cold winter months. When spring arrives, the ice starts to melt.
\newline
S
S
S
models the ice sheet's thickness (in meters) after
t
t
t
weeks.
\newline
S
=
−
0.25
t
+
4
S=-0.25 t+4
S
=
−
0.25
t
+
4
\newline
What is the ice sheet's thickness at the beginning of spring?
\newline
meters
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At Simba Travel Agency, the price of a climbing trip to Mount Kilimanjaro includes an initial fee plus a constant fee per meter.
\newline
F
F
F
represents the fee (in dollars) for climbing
d
d
d
meters.
\newline
F
=
110
+
0.12
d
F=110+0.12 d
F
=
110
+
0.12
d
\newline
What is the increase in price for every
100
100
100
meters climbed?
\newline
$
\$
$
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Rachel is a stunt driver, and she's escaping from a building that is about to explode!
\newline
D
D
D
represents Rachel's remaining distance (in meters) after
t
t
t
seconds.
\newline
D
=
−
38
t
+
220
D=-38 t+220
D
=
−
38
t
+
220
\newline
How long is the distance Rachel has to drive?
\newline
meters
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A big ship drops its anchor.
\newline
E
E
E
represents the anchor's elevation relative to the water's surface (in meters) as a function of time
t
t
t
(in seconds).
\newline
E
=
−
2.4
t
+
75
E=-2.4 t+75
E
=
−
2.4
t
+
75
\newline
How far does the anchor drop every
5
5
5
seconds?
\newline
meters
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Mr. Mole left his burrow and started digging his way down.
\newline
A
A
A
represents Mr. Mole's altitude relative to the ground (in meters) after
t
t
t
minutes.
\newline
A
=
−
2.3
t
−
7
A=-2.3 t-7
A
=
−
2.3
t
−
7
\newline
How fast did Mr. Mole descend?
\newline
meters per minute
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A young sumo wrestler decided to go on a special diet to gain weight rapidly.
\newline
W
W
W
represents the wrestler's weight (in kilograms) as a function of time
t
t
t
(in months).
\newline
W
=
80
+
5.4
t
W=80+5.4 t
W
=
80
+
5.4
t
\newline
How much weight does the wrestler gain every
2
2
2
months?
\newline
kilograms
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D
=
−
38
t
+
220
D=-38 t+220
D
=
−
38
t
+
220
\newline
Rachel is driving a race car at a constant speed on a closed course. The formula shown describes the remaining distance,
D
D
D
, measured in meters, that Rachel has to travel after
t
t
t
seconds. What is Rachel's speed in meters per second?
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R
=
42
−
0.7
t
R=42-0.7 t
R
=
42
−
0.7
t
\newline
Quinn returned home one summer's day to find it extremely hot. He turned the air conditioner on, and the room's temperature began decreasing at a constant rate. The given equation shown gives the room's temperature,
R
R
R
, in degrees Celsius,
t
t
t
minutes after Quinn turned on the air conditioner. What was the room's temperature, in degrees Celsius, when Quinn returned home?
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T
=
235
+
980
d
T=235+980 d
T
=
235
+
980
d
\newline
The torque,
T
T
T
, in newton-meters, on a diving board with a
100
100
100
kilogram weight placed
d
d
d
meters from the diving board's fulcrum is given by the equation. How much torque in newton-meters is on the diving board when the weight is placed at the fulcrum?
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A
=
2450
+
740
s
A=2450+740 s
A
=
2450
+
740
s
\newline
The total amount,
A
A
A
, in thousands of dollars, of scholarship money given out by a college throughout its history at a time of
s
s
s
semesters after the summer of the year
2000
2000
2000
is given by the equation. What was the total amount of scholarship money, in thousands of dollars, given out by the college up to the summer of the year
2000
2000
2000
?
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Kevin invested
$
1
,
000
\$1,000
$1
,
000
in a savings account with a daily interest rate of
0.1
%
0.1\%
0.1%
. Write an exponential function to represent the total amount,
s
(
x
)
s(x)
s
(
x
)
, in his account after
x
x
x
days.
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Sally purchased a laptop for `$800` and has to pay a service fee of `$20` for each day until it is delivered. Write a linear function to represent the total cost,
h
(
x
)
h(x)
h
(
x
)
, after
x
x
x
days.
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Bob rented a car for a flat fee of `$150` and an additional `$10` per day. Write a linear function that represents the total cost,
f
(
x
)
f(x)
f
(
x
)
, after
x
x
x
days.
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Raymond works for a pharmaceutical company that is testing the effectiveness of a new medication. He gave one of the study participants
400
400
400
milligrams of the medication, and he knows the amount remaining in the participant's body will decrease by a factor of
1
/
3
1/3
1/3
each hour.
\newline
Write an exponential equation in the form
y
=
a
(
b
)
x
y = a(b)^x
y
=
a
(
b
)
x
that can model the amount of medication,
y
y
y
, remaining in the participant's body after
x
x
x
hours.
\newline
Use whole numbers, decimals, or simplified fractions for the values of
a
a
a
and
b
b
b
.
\newline
y
=
y =
y
=
______
\newline
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