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Math Problems
Algebra 2
Interpret the exponential function word problem
On the first day of spring, an entire field of flowering trees blossoms. The population of locusts consuming these flowers rapidly increases as the trees blossom.
\newline
The relationship between the elapsed time,
t
t
t
, in weeks, since the beginning of spring, and the total number of locusts,
N
(
t
)
N(t)
N
(
t
)
, is modeled by the following function:
\newline
N
(
t
)
=
300
⋅
9
t
N(t)=300 \cdot 9^{t}
N
(
t
)
=
300
⋅
9
t
\newline
Complete the following sentence about the rate of change in the locust population.
\newline
Round your answer to two decimal places.
\newline
The number of locusts is tripled every weeks.
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The country of Freedonia has decided to reduce its carbondioxide emission by
35
%
35 \%
35%
each year. This year the country emitted
40
40
40
million tons of carbon-dioxide.
\newline
Write a function that gives Freedonia's carbon-dioxide emissions in million tons,
E
(
t
)
,
t
E(t), t
E
(
t
)
,
t
years from today.
\newline
E
(
t
)
=
□
E(t)=\square
E
(
t
)
=
□
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Carbon
−
14
-14
−
14
is an element that loses about
10
%
10 \%
10%
of its mass every millennium (i.e.,
1000
1000
1000
years). A sample of Carbon
−
14
-14
−
14
has
600
600
600
grams.
\newline
Write a function that gives the sample's mass in grams,
S
(
t
)
,
t
S(t), t
S
(
t
)
,
t
millennia from today.
\newline
S
(
t
)
=
□
S(t)=\square
S
(
t
)
=
□
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The country of Sylvania has decided to reduce the number of its illiterate citizens by
3
5
\frac{3}{5}
5
3
each year. This year there are
9000
9000
9000
illiterate people in the country.
\newline
Write a function that gives the number of illiterate people in Sylvania,
P
(
t
)
P(t)
P
(
t
)
,
t
t
t
years from today.
\newline
P
(
t
)
=
□
P(t)=\square
P
(
t
)
=
□
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Einsteinium
−
253
-253
−
253
is an element that loses about
2
3
\frac{2}{3}
3
2
of its mass every month. A sample of Einsteinium
253
253
253
has
450
450
450
grams.
\newline
Write a function that gives the sample's mass in grams,
S
(
t
)
,
t
S(t), t
S
(
t
)
,
t
months from today.
\newline
S
(
t
)
=
□
S(t)=\square
S
(
t
)
=
□
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At the moment a certain medicine is injected, its concentration in the bloodstream is
120
120
120
milligrams per liter. From that moment forward, the medicine's concentration drops by
30
%
30 \%
30%
each hour.
\newline
Write a function that gives the medicine's concentration in milligrams per liter,
C
(
t
)
C(t)
C
(
t
)
,
t
t
t
hours after the medicine was injected.
\newline
C
(
t
)
=
□
C(t)=\square
C
(
t
)
=
□
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A computer sells for
$
900
\$ 900
$900
and loses
30
%
30 \%
30%
of its value per year.
\newline
Write a function that gives the computer's value,
V
(
t
)
,
t
V(t), t
V
(
t
)
,
t
years after it is sold.
\newline
V
(
t
)
=
□
V(t)=\square
V
(
t
)
=
□
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A car sells for
$
5000
\$ 5000
$5000
and loses
1
10
\frac{1}{10}
10
1
of its value each year.
\newline
Write a function that gives the car's value,
V
(
t
)
,
t
V(t), t
V
(
t
)
,
t
years after it is sold.
\newline
V
(
t
)
=
□
V(t)= \square
V
(
t
)
=
□
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Nana has a water purifier that filters
1
3
\frac{1}{3}
3
1
of the contaminants each hour. She used it to purify water that had
1
2
\frac{1}{2}
2
1
kilogram of contaminants.
\newline
Write a function that gives the remaining amount of contaminants in kilograms,
C
(
t
)
,
t
C(t), t
C
(
t
)
,
t
hours after Nana started purifying the water.
\newline
C
(
t
)
=
□
C(t)= \square
C
(
t
)
=
□
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At the moment a hot cake is put in a cooler, the difference between the cake's and the cooler's temperatures is
5
0
∘
50^{\circ}
5
0
∘
Celsius. This causes the cake to cool and the temperature difference loses
1
5
\frac{1}{5}
5
1
of its value every minute.
\newline
Write a function that gives the temperature difference in degrees Celsius,
D
(
t
)
,
t
D(t), t
D
(
t
)
,
t
minutes after the cake was put in the cooler.
\newline
D
(
t
)
=
□
D(t)=\square
D
(
t
)
=
□
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A forest has
800
800
800
pine trees, but a disease is introduced that kills
1
4
\frac{1}{4}
4
1
of the pine trees in the forest every year.
\newline
Write a function that gives the number of pine trees remaining
P
(
t
)
P(t)
P
(
t
)
in the forest
t
t
t
years after the disease is introduced.
\newline
P
(
t
)
=
□
P(t)=\square
P
(
t
)
=
□
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The lion population in a certain reserve drops by
5
%
5 \%
5%
every year. Currently, the population's size is
200
200
200
.
\newline
Write a function that gives the lion population size,
P
(
t
)
P(t)
P
(
t
)
,
t
t
t
years from today.
\newline
P
(
t
)
=
□
P(t)=\square
P
(
t
)
=
□
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Does the function model exponential growth or decay?
\newline
h
(
t
)
=
0.25
⋅
1.
5
t
h(t)=0.25 \cdot 1.5^{t}
h
(
t
)
=
0.25
⋅
1.
5
t
\newline
Choose
1
1
1
answer:
\newline
(A) Growth
\newline
(B) Decay
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Does the function model exponential growth or decay?
\newline
g
(
x
)
=
1
3
⋅
(
2
9
)
x
g(x)=\frac{1}{3} \cdot\left(\frac{2}{9}\right)^{x}
g
(
x
)
=
3
1
⋅
(
9
2
)
x
\newline
Choose
1
1
1
answer:
\newline
(A) Growth
\newline
(B) Decay
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Does the function model exponential growth or decay?
\newline
h
(
x
)
=
3
4
⋅
(
1
6
)
x
h(x)=\frac{3}{4} \cdot\left(\frac{1}{6}\right)^{x}
h
(
x
)
=
4
3
⋅
(
6
1
)
x
\newline
Choose
1
1
1
answer:
\newline
(A) Growth
\newline
(B) Decay
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Does the function model exponential growth or decay?
\newline
g
(
x
)
=
7
3
⋅
2
x
g(x)=\frac{7}{3} \cdot 2^{x}
g
(
x
)
=
3
7
⋅
2
x
\newline
Choose
1
1
1
answer:
\newline
(A) Growth
\newline
(B) Decay
Get tutor help
Does the function model exponential growth or decay?
\newline
g
(
t
)
=
1.7
⋅
0.
8
t
g(t)=1.7 \cdot 0.8^{t}
g
(
t
)
=
1.7
⋅
0.
8
t
\newline
Choose
1
1
1
answer:
\newline
(A) Growth
\newline
(B) Decay
Get tutor help
Does the function model exponential growth or decay?
\newline
f
(
t
)
=
5
⋅
(
3
7
)
t
f(t)=5 \cdot\left(\frac{3}{7}\right)^{t}
f
(
t
)
=
5
⋅
(
7
3
)
t
\newline
Choose
1
1
1
answer:
\newline
(A) Growth
\newline
(B) Decay
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Does the function model exponential growth or decay?
\newline
f
(
t
)
=
1
4
⋅
4
t
f(t)=\frac{1}{4} \cdot 4^{t}
f
(
t
)
=
4
1
⋅
4
t
\newline
Choose
1
1
1
answer:
\newline
(A) Growth
\newline
(B) Decay
Get tutor help
Does the function model exponential growth or decay?
\newline
f
(
x
)
=
3
⋅
(
7
4
)
x
f(x)=3 \cdot\left(\frac{7}{4}\right)^{x}
f
(
x
)
=
3
⋅
(
4
7
)
x
\newline
Choose
1
1
1
answer:
\newline
(A) Growth
\newline
(B) Decay
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At the moment a hot iron rod is plunged into freezing water, the difference between the rod's and the water's temperatures is
10
0
∘
100^{\circ}
10
0
∘
Celsius. This causes the iron to cool and the temperature difference drops by
60
%
60 \%
60%
every second.
\newline
Write a function that gives the temperature difference in degrees Celsius,
D
(
t
)
,
t
D(t), t
D
(
t
)
,
t
seconds after the rod was plunged into the water.
\newline
D
(
t
)
=
□
D(t)=\square
D
(
t
)
=
□
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Shota invests
$
2000
\$ 2000
$2000
in a certificate of deposit that earns
2
%
2 \%
2%
in interest each year.
\newline
Write a function that gives the total value
V
(
t
)
V(t)
V
(
t
)
, in dollars, of the investment
t
t
t
years from now.
\newline
Do not enter commas in your answer.
\newline
V
(
t
)
=
V(t)=
V
(
t
)
=
□
\square
□
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The expression
1350
(
1.05
)
t
1350(1.05)^{t}
1350
(
1.05
)
t
models the average wages, in dollars, in the US as a function of the number of years since
1930
1930
1930
.
\newline
What does
1
1
1
.
05
05
05
represent in this expression?
\newline
Choose
1
1
1
answer:
\newline
(A) The average wages double every
1
1
1
.
05
05
05
years.
\newline
(B) The average wages in the US were about
$
1.05
\$ 1.05
$1.05
in
1930
1930
1930
.
\newline
(C) The average wages in the US increase by about
5
%
5 \%
5%
each year.
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The expression
3
(
1.5
)
t
3(1.5)^{t}
3
(
1.5
)
t
models the number of bacteria in a culture as a function of the number of hours since the culture was created.
\newline
What does
3
3
3
represent in this expression?
\newline
Choose
1
1
1
answer:
\newline
(A) There were initially
3
3
3
bacteria in the culture.
\newline
(B) The culture was created
3
3
3
hours ago.
\newline
(C) The number of bacteria is multiplied by
3
3
3
each hour.
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The expression
5
⋅
1.1
5
t
5 \cdot 1.15^{t}
5
⋅
1.1
5
t
gives the area, in square meters, damaged by a fire as a function of minutes since the fire was discovered.
\newline
What does
5
5
5
represent in this expression?
\newline
Choose
1
1
1
answer:
\newline
(A) An area of
5
5
5
square meters had been damaged by the fire by the time it was discovered.
\newline
(B) It has been
5
5
5
minutes since the fire was discovered.
\newline
(C) The area damaged by the fire is multiplied by
5
5
5
each minute.
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The expression
4
⋅
3
t
4 \cdot 3^{t}
4
⋅
3
t
gives the number of ducks living in Anoop's pond
t
t
t
years after he built it.
\newline
What does
4
4
4
represent in this expression?
\newline
Choose
1
1
1
answer:
\newline
(A) It has been
4
4
4
years since Anoop built the pond.
\newline
(B) The number of ducks living in Anoop's pond is multiplied by
4
4
4
every year.
\newline
(C) There were
4
4
4
ducks living in Anoop's pond when he first built it.
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Javier bought a painting for
$
150
\$ 150
$150
. Each year, the painting's value increases by a factor of
1
1
1
.
15
15
15
.
\newline
Which expression gives the painting's value after
7
7
7
years?
\newline
Choose
1
1
1
answer:
\newline
(A)
150
+
1.15
⋅
7
150+1.15 \cdot 7
150
+
1.15
⋅
7
\newline
(B)
150
⋅
1.1
5
7
150 \cdot 1.15^{7}
150
⋅
1.1
5
7
\newline
(C)
(
150
⋅
1.15
)
7
(150 \cdot 1.15)^{7}
(
150
⋅
1.15
)
7
\newline
(D)
(
150
+
1.15
)
⋅
7
(150+1.15) \cdot 7
(
150
+
1.15
)
⋅
7
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Karim sent a chain email to
10
10
10
of his friends. The number of people who got the email increases by a factor of
1
1
1
.
4
4
4
every week.
\newline
Which expression gives the number of people who got the email after
6
6
6
weeks?
\newline
Choose
1
1
1
answer:
\newline
(A)
10
+
1.4
⋅
6
10+1.4 \cdot 6
10
+
1.4
⋅
6
\newline
(B)
10
⋅
1.
4
6
10 \cdot 1.4^{6}
10
⋅
1.
4
6
\newline
(C)
(
10
⋅
1.4
)
6
(10 \cdot 1.4)^{6}
(
10
⋅
1.4
)
6
\newline
(D)
(
10
+
1.4
)
⋅
6
(10+1.4) \cdot 6
(
10
+
1.4
)
⋅
6
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Moore's law says that the number of transistors in a dense integrated circuit increases by
41
%
41 \%
41%
every year. In
1974
1974
1974
, a dense integrated circuit was produced with
5000
5000
5000
transistors.
\newline
Which expression gives the number of transistors in a dense integrated circuit in
1979
1979
1979
?
\newline
Choose
1
1
1
answer:
\newline
(A)
5000
+
0.4
1
5
5000+0.41^{5}
5000
+
0.4
1
5
\newline
(B)
5000
⋅
0.4
1
5
5000 \cdot 0.41^{5}
5000
⋅
0.4
1
5
\newline
(C)
5000
(
1
+
0.41
)
5
5000(1+0.41)^{5}
5000
(
1
+
0.41
)
5
\newline
(D)
5000
+
(
1
+
0.41
)
5
5000+(1+0.41)^{5}
5000
+
(
1
+
0.41
)
5
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In
1950
1950
1950
, the per capita gross domestic product (GDP) of Australia was approximately
$
1800
\$ 1800
$1800
. Each year afterwards, the per capita GDP increased by approximately
6.7
%
6.7 \%
6.7%
.
\newline
Write a function that gives the approximate per capita GDP
G
(
t
)
G(t)
G
(
t
)
of Australia
t
t
t
years after
1950
1950
1950
. Do not enter commas in your answer.
\newline
G
(
t
)
=
G(t)=
G
(
t
)
=
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Ngozi earns
$
24
,
000
\$ 24,000
$24
,
000
in salary in the first year she works as an interpreter. Each year, she earns a
3.5
%
3.5 \%
3.5%
raise.
\newline
Write a function that gives Ngozi's salary
S
(
t
)
S(t)
S
(
t
)
, in dollars,
t
t
t
years after she starts to work as an interpreter. Do not enter commas in your answer.
\newline
S
(
t
)
=
S(t)=
S
(
t
)
=
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The town of Madison has a population of
25
25
25
,
000
000
000
. The population is increasing by a factor of
1
1
1
.
12
12
12
each year.
\newline
Write a function that gives the population
P
(
t
)
P(t)
P
(
t
)
in Madison
t
t
t
years from now.
\newline
Do not use commas in your answer.
\newline
P
(
t
)
=
P(t)=
P
(
t
)
=
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On the day that a certain celebrity proposes marriage,
10
10
10
people know about it. Each day afterward, the number of people who know triples.
\newline
Write a function that gives the total number
n
(
t
)
n(t)
n
(
t
)
of people who know about the celebrity proposing
t
t
t
days after the celebrity proposes.
\newline
n
(
t
)
=
n(t)=
n
(
t
)
=
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Mapiya writes a series of novels. She earned
$
75
,
000
\$ 75,000
$75
,
000
for the first book, and her cumulative earnings double with each sequel that she writes.
\newline
Write a function that gives Mapiya's cumulative earnings
E
(
n
)
E(n)
E
(
n
)
, in dollars, when she has written
n
n
n
sequels.
\newline
Do not enter commas in your answer.
\newline
E
(
n
)
=
E(n)=
E
(
n
)
=
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When the Dragons have lost their most recent game,
200
200
200
fans buy tickets. For each consecutive win the Dragons have, the number of tickets fans buy increases by a factor of
1
1
1
.
1
1
1
.
\newline
Write a function that gives the number of tickets
t
(
w
)
t(w)
t
(
w
)
fans buy to Dragons games after the Dragons have had
w
w
w
consecutive wins.
\newline
t
(
w
)
=
□
t(w)=\square
t
(
w
)
=
□
Get tutor help
The number of nano-related patents that are granted in the US increases by a factor of
1
1
1
.
2
2
2
every year. In
1991
1991
1991
, there were
60
60
60
nanorelated patents.
\newline
Which expression gives the number of patents in
1998
1998
1998
?
\newline
Choose
1
1
1
answer:
\newline
(A)
60
⋅
1.
2
7
60 \cdot 1.2^{7}
60
⋅
1.
2
7
\newline
(B)
60
+
(
1
+
1.2
)
7
60+(1+1.2)^{7}
60
+
(
1
+
1.2
)
7
\newline
(C)
60
⋅
(
1
+
1.2
)
7
60 \cdot(1+1.2)^{7}
60
⋅
(
1
+
1.2
)
7
\newline
(D)
60
+
1.
2
7
60+1.2^{7}
60
+
1.
2
7
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Shota invests
$
1000
\$ 1000
$1000
in a certificate of deposit that earns interest. The investment's value is multiplied by
1
1
1
.
02
02
02
each year.
\newline
Which expression gives the investment's value after
5
5
5
years?
\newline
Choose
1
1
1
answer:
\newline
(A)
1000
+
1.0
2
5
1000+1.02^{5}
1000
+
1.0
2
5
\newline
(B)
1000
+
(
1
+
1.02
)
5
1000+(1+1.02)^{5}
1000
+
(
1
+
1.02
)
5
\newline
(C)
1000
⋅
1.0
2
5
1000 \cdot 1.02^{5}
1000
⋅
1.0
2
5
\newline
(D)
1000
⋅
(
1
+
1.02
)
5
1000 \cdot(1+1.02)^{5}
1000
⋅
(
1
+
1.02
)
5
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The town of Springfield, USA has a population of
9900
9900
9900
people. The population is growing at a rate of
2.3
%
2.3 \%
2.3%
each year.
\newline
Which expression gives the town's population
5
5
5
years from now?
\newline
Choose
1
1
1
answer:
\newline
(A)
9900
⋅
0.02
3
5
9900 \cdot 0.023^{5}
9900
⋅
0.02
3
5
\newline
(B)
9900
+
0.02
3
5
9900+0.023^{5}
9900
+
0.02
3
5
\newline
(C)
9900
(
1
+
0.023
)
5
9900(1+0.023)^{5}
9900
(
1
+
0.023
)
5
\newline
(D)
9900
+
(
1
+
0.023
)
5
9900+(1+0.023)^{5}
9900
+
(
1
+
0.023
)
5
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The number of bacteria in a petri dish is multiplied by a factor of
1
1
1
.
2
2
2
every hour. There were initially
500
500
500
bacteria.
\newline
Which expression gives the number of bacteria after
3
3
3
hours?
\newline
Choose
1
1
1
answer:
\newline
(A)
500
+
1.2
⋅
1.2
⋅
1.2
500+1.2 \cdot 1.2 \cdot 1.2
500
+
1.2
⋅
1.2
⋅
1.2
\newline
(B)
500
⋅
1.2
⋅
1.2
⋅
1.2
500 \cdot 1.2 \cdot 1.2 \cdot 1.2
500
⋅
1.2
⋅
1.2
⋅
1.2
\newline
(C)
500
+
1.2
+
1.2
+
1.2
500+1.2+1.2+1.2
500
+
1.2
+
1.2
+
1.2
\newline
(D)
500
⋅
(
1.2
+
1.2
+
1.2
)
500 \cdot(1.2+1.2+1.2)
500
⋅
(
1.2
+
1.2
+
1.2
)
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Fidel has a rare coin worth
$
550
\$ 550
$550
. Each decade, the coin's value increases by
10
%
10 \%
10%
.
\newline
Which expression gives the coin's value,
6
6
6
decades from now?
\newline
Choose
1
1
1
answer:
\newline
(A)
550
⋅
0.
1
6
550 \cdot 0.1^{6}
550
⋅
0.
1
6
\newline
(B)
550
(
1
+
0.1
)
6
550(1+0.1)^{6}
550
(
1
+
0.1
)
6
\newline
(C)
550
+
(
1
+
0.1
)
6
550+(1+0.1)^{6}
550
+
(
1
+
0.1
)
6
\newline
(D)
550
+
0.
1
6
550+0.1^{6}
550
+
0.
1
6
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Eric made two investments:
\newline
- Investment
Q
\mathrm{Q}
Q
has a value of
$
500
\$ 500
$500
at the end of the first year and increases by
$
45
\$ 45
$45
per year.
\newline
- Investment
R
\mathrm{R}
R
has a value of
$
400
\$ 400
$400
at the end of the first year and increases by
10
%
10 \%
10%
per year.
\newline
Eric checks the value of his investments once a year, at the end of the year.
\newline
What is the first year in which Eric sees that investment
R
R
R
's value exceeded investment Q's value?
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The expression
1.5
⋅
1.0
9
t
1.5 \cdot 1.09^{t}
1.5
⋅
1.0
9
t
models the housing costs, in thousands of dollars, for summer term
t
t
t
years since Chinedu applied to his college.
\newline
What does
1
1
1
.
09
09
09
represent in this expression?
\newline
Choose
1
1
1
answer:
\newline
(A) The summer housing costs were
$
1
,
090
\$ 1,090
$1
,
090
the year that Chinedu applied to his college.
\newline
(B) Chinedu applied to his school
9
9
9
years ago.
\newline
(C) The summer housing costs increase by
9
%
9 \%
9%
each year.
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Every year
39
39
39
million cars cross the Golden Gate Bridge in San Francisco.
\newline
Which kind of function best models the relationship between time and the cumulative number of cars that have crossed the Golden Gate Bridge?
\newline
Choose
1
1
1
answer:
\newline
(A) Linear
\newline
(B) Exponential
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Fidel has a rare coin worth
$
550
\$ 550
$550
. Each decade, the coin's value increases by
10
%
10 \%
10%
.
\newline
Which kind of function best models the relationship between time and the coin's value?
\newline
Choose
1
1
1
answer:
\newline
(A) Linear
\newline
(B) Exponential
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Enrico deposited
$
2000
\$ 2000
$2000
in a savings account. Each month he will deposit an additional
$
25
\$ 25
$25
.
\newline
Which kind of function best models the relationship between time and the total amount in the savings account?
\newline
Choose
1
1
1
answer:
\newline
(A) Linear
\newline
(B) Exponential
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The number of visitors to the Eiffel Tower in Paris increases by
1.5
%
1.5\%
1.5%
each year.
\newline
Which kind of function best models the relationship between time and the number of visitors to the Eiffel Tower?
\newline
Choose
1
1
1
answer:
\newline
(A) Linear
\newline
(B) Exponential
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