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Math Problems
Algebra 2
Ratio and Quadratic equation
A certain companys main source of income is a mobile app. The company's annual profit (in millions of dollars) as a function of the app's price (In dollars) is modeled by
P
(
x
)
=
−
2
(
x
−
3
)
(
x
−
11
)
P(x) = -2(x-3)(x-11)
P
(
x
)
=
−
2
(
x
−
3
)
(
x
−
11
)
Which app prices will result in
$
0
\$0
$0
annual profit
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Square A has side lengths that are
166
166
166
times the side lengths of square B. The area of square A is
k
k
k
times the area of square B. What is the value of
k
k
k
?
\newline
A)
166
166
166
\newline
B)
332
332
332
\newline
C)
16600
16600
16600
\newline
D)
27556
27556
27556
\newline
E)
27556
27556
27556
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Simon has
meters
\text{meters}
meters
of fencing to build a rectangular garden. The garden's area (in square meters) as a function of the garden's width
(in meters)
\text{(in meters)}
(in meters)
is modeled by What width will produce the maximum garden area?
meters
\text{meters}
meters
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Simon has
meters
\text{meters}
meters
of fencing to build a rectangular garden. The garden's area (in square meters) as a function of the garden's width
(in meters)
\text{(in meters)}
(in meters)
is modeled by What width will produce the maximum garden area?
meters
\text{meters}
meters
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Simon has
meters
\text{meters}
meters
of fencing to build a rectangular garden. The garden's area (in square meters) as a function of the garden's width
(in meters)
\text{(in meters)}
(in meters)
is modeled by What width will produce the maximum garden area?
meters
\text{meters}
meters
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An isosceles triangle has a height of
12.5
m
12.5 \mathrm{~m}
12.5
m
and two equal angles that measure
5
5
∘
55^{\circ}
5
5
∘
. Determine the area of the triangle.
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\newline
The price value,
V
V
V
, of a car that is
t
t
t
years old is given by
V
=
f
(
t
)
=
17000
−
3100
t
V=f(t)=17000-3100 t
V
=
f
(
t
)
=
17000
−
3100
t
. Find thr domain and range of
f
(
t
)
f(t)
f
(
t
)
.
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A
300
300
300
-room hotel collects
$
75
\$75
$75
per occupied room and does not collect any money for vacant rooms. Which of the following functions best represents how many dollars,
d
d
d
, the hotel generates if there are
v
v
v
vacant rooms in the hotel?
\newline
Choose
1
1
1
answer:
\newline
(A)
d
=
75
(
300
−
v
)
d=75(300-v)
d
=
75
(
300
−
v
)
\newline
(B)
d
=
75
(
300
+
v
)
d=75(300+v)
d
=
75
(
300
+
v
)
\newline
(C)
d
=
300
(
75
−
v
)
d=300(75-v)
d
=
300
(
75
−
v
)
\newline
(D)
d
=
300
(
75
+
v
)
d=300(75+v)
d
=
300
(
75
+
v
)
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An object is launched from a platform.
\newline
Its height (in meters),
x
x
x
seconds after the launch, is modeled by
\newline
h
(
x
)
=
−
5
(
x
+
1
)
(
x
−
9
)
h(x)=-5(x+1)(x-9)
h
(
x
)
=
−
5
(
x
+
1
)
(
x
−
9
)
\newline
How many seconds after launch will the object hit the ground?
\newline
□
\square
□
seconds
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Simon has
160
160
160
meters of fencing to build a rectangular garden.
\newline
The garden's area (in square meters) as a function of the garden's width
x
x
x
(in meters) is modeled by
\newline
A
(
x
)
=
−
x
(
x
−
80
)
A(x)=-x(x-80)
A
(
x
)
=
−
x
(
x
−
80
)
\newline
What is the maximum area possible?
\newline
□
\square
□
square meters
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The power generated by an electrical circuit (in watts) as a function of its current
\newline
x
x
x
(in amperes) is modeled by
\newline
P
(
x
)
=
−
15
x
(
x
−
8
)
P(x)=-15 x(x-8)
P
(
x
)
=
−
15
x
(
x
−
8
)
\newline
What current will produce the maximum power?
\newline
□
\square
□
amperes
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The derivative of a function
g
g
g
is given by
\newline
g
′
(
x
)
=
x
3
−
2
x
2
−
3
x
+
4.
g^{\prime}(x)=x^{3}-2 x^{2}-3 x+4 .
g
′
(
x
)
=
x
3
−
2
x
2
−
3
x
+
4.
\newline
On which intervals is the graph of
g
g
g
increasing?
\newline
Use a graphing calculator.
\newline
Choose
1
1
1
answer:
\newline
(A)
x
≤
−
1.562
x \leq-1.562
x
≤
−
1.562
and
\newline
1
≤
x
≤
2.562
1 \leq x \leq 2.562
1
≤
x
≤
2.562
\newline
(B)
−
1.562
≤
x
≤
1
-1.562 \leq x \leq 1
−
1.562
≤
x
≤
1
and
x
≥
2.562
x \geq 2.562
x
≥
2.562
\newline
(C)
x
≤
−
0.535
x \leq-0.535
x
≤
−
0.535
and
x
≥
1.869
x \geq 1.869
x
≥
1.869
\newline
(D)
−
0.535
≤
x
≤
1.869
-0.535 \leq x \leq 1.869
−
0.535
≤
x
≤
1.869
\newline
(E) All real numbers
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The radius of the base of a cylinder is increasing at a rate of
7
7
7
millimeters per hour.
\newline
The height of the cylinder is fixed at
1
1
1
.
5
5
5
millimeters.
\newline
At a certain instant, the radius is
12
12
12
millimeters.
\newline
What is the rate of change of the volume of the cylinder at that instant (in cubic millimeters per hour)?
\newline
Choose
1
1
1
answer:
\newline
(A)
216
π
216 \pi
216
π
\newline
(B)
1512
π
1512 \pi
1512
π
\newline
(C)
126
π
126 \pi
126
π
\newline
(D)
252
π
252 \pi
252
π
\newline
The volume of a cylinder with radius
r
r
r
and height
h
h
h
is
π
r
2
h
\pi r^{2} h
π
r
2
h
.
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The radius of the base of a cylinder is increasing at a rate of
7
7
7
millimeters per hour.
\newline
The height of the cylinder is fixed at
1
1
1
.
5
5
5
millimeters.
\newline
At a certain instant, the radius is
12
12
12
millimeters.
\newline
What is the rate of change of the volume of the cylinder at that instant (in cubic millimeters per hour)?
\newline
Choose
1
1
1
answer:
\newline
(A)
252
π
252 \pi
252
π
\newline
(B)
216
π
216 \pi
216
π
\newline
(C)
1512
π
1512 \pi
1512
π
\newline
(D)
126
π
126 \pi
126
π
\newline
The volume of a cylinder with radius
r
r
r
and height
h
h
h
is
π
r
2
h
\pi r^{2} h
π
r
2
h
.
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R
=
206.835
−
1.015
W
−
84.6
S
R=206.835-1.015 W-84.6 S
R
=
206.835
−
1.015
W
−
84.6
S
\newline
The formula gives the Flesch Reading Ease score
R
R
R
for a passage of text with an average of
W
W
W
words per sentence and an average of
S
S
S
syllables per word. Which of the following equations correctly shows the average number of syllables per word in terms of the average number of words per sentence and the Flesch Reading Ease score?
\newline
Choose
1
1
1
answer:
\newline
(A)
S
=
R
84.6
+
206.835
−
1.015
W
S=\frac{R}{84.6}+206.835-1.015 W
S
=
84.6
R
+
206.835
−
1.015
W
\newline
(B)
S
=
R
−
84.6
−
206.835
+
1.015
W
S=\frac{R}{-84.6}-206.835+1.015 W
S
=
−
84.6
R
−
206.835
+
1.015
W
\newline
(C)
S
=
R
84.6
−
206.835
+
1.015
W
S=\frac{R}{84.6-206.835+1.015 W}
S
=
84.6
−
206.835
+
1.015
W
R
\newline
(D)
S
=
R
−
206.835
+
1.015
W
−
84.6
S=\frac{R-206.835+1.015 W}{-84.6}
S
=
−
84.6
R
−
206.835
+
1.015
W
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An object is launched from a platform.
\newline
Its height (in meters),
x
x
x
seconds after the launch, is modeled by
\newline
h
(
x
)
=
−
5
(
x
+
1
)
(
x
−
9
)
h(x)=-5(x+1)(x-9)
h
(
x
)
=
−
5
(
x
+
1
)
(
x
−
9
)
\newline
What is the height of the object at the time of launch?
\newline
□
\square
□
meters
\text{meters}
meters
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In San Diego, the number of daylight hours is dependent on the day of the year. The number of daylight hours
D
D
D
can be modeled by the sinusoidal function
D
(
t
)
=
2.4
sin
(
0.017
(
t
−
81
)
)
+
12
D(t)=2.4 \sin (0.017(t-81))+12
D
(
t
)
=
2.4
sin
(
0.017
(
t
−
81
))
+
12
, where
t
t
t
is the day of the year from
t
=
1
t=1
t
=
1
to
t
=
365
t=365
t
=
365
. Based on this model, which of the following statements is correct?
\newline
(A) The minimum number of daylight hours in San Diego is
12
12
12
hours.
\newline
(B) The maximum number of daylight hours in San Diego is
12
12
12
hours.
\newline
(C) The minimum number of daylight hours in San Diego is
9
9
9
.
6
6
6
hours.
\newline
(D) The maximum number of daylight hours in San Diego is
9
9
9
.
6
6
6
hours.
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Our school bought
300
300
300
Pencils for
$
75
\$75
$75
. They came in boxes of
12
12
12
Pencils. How much did each box cost?
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Simon has
160
160
160
meters of fencing to build a rectangular garden.
\newline
The garden's area (in square meters) as a function of the garden's width
x
x
x
(in meters) is modeled by
\newline
A
(
x
)
=
−
x
(
x
−
80
)
A(x)=-x(x-80)
A
(
x
)
=
−
x
(
x
−
80
)
\newline
What width will produce the maximum garden area?
\newline
□
\square
□
meters
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Δ
P
=
6.2
×
(
600
(
450000
)
1
/
3
)
1.1
\Delta P=6.2\times(600(450000)^{1/3})^{1.1}
Δ
P
=
6.2
×
(
600
(
450000
)
1/3
)
1.1
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Mr. Kirby's physics class went to the football field to launch a rocket into the air. The rocket's height in feet,
f
(
x
)
f(x)
f
(
x
)
, depends on the number of seconds,
x
x
x
, that have elapsed since launch.
\newline
What does
f
(
8
)
=
0
f(8) = 0
f
(
8
)
=
0
tell you?
\newline
Choices:
\newline
(A)The rocket starts its launch
8
8
8
feet above the ground.
\newline
(B)After
8
8
8
seconds, the rocket has already landed on the field.
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The fish population in a certain part of the ocean (in thousands of fish) as a function of the water's temperature (in degrees Celsius) is modeled by:
p
(
x
)
=
−
2
x
2
+
40
x
−
72
p(x)=-2x^2+40x-72
p
(
x
)
=
−
2
x
2
+
40
x
−
72
Which temperatures will result in no fish (i.e. population)?
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Jason used a discount coupon for
5
%
5 \%
5%
off at his favorite clothing store. He bought
9
9
9
pairs of shoes for
x
x
x
dollars each and
7
7
7
coats for
y
y
y
dollars each. Which expression represents the total amount Jason paid after the coupon was applied?
\newline
5
(
9
x
+
7
y
)
5(9 x+7 y)
5
(
9
x
+
7
y
)
\newline
0.05
(
9
x
+
7
y
)
+
(
9
x
+
7
y
)
0.05(9 x+7 y)+(9 x+7 y)
0.05
(
9
x
+
7
y
)
+
(
9
x
+
7
y
)
\newline
0.95
(
9
x
+
7
y
)
0.95(9 x+7 y)
0.95
(
9
x
+
7
y
)
\newline
(
9
x
+
7
y
)
−
0.95
(
9
x
+
7
y
)
(9 x+7 y)-0.95(9 x+7 y)
(
9
x
+
7
y
)
−
0.95
(
9
x
+
7
y
)
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Magan bought
f
f
f
folders that cost
$
0.30
\$ 0.30
$0.30
each and
m
m
m
markers that cost
$
0.65
\$ 0.65
$0.65
each. Sales tax is
8.25
%
8.25 \%
8.25%
. Which expression represents the total amount Magan paid, including tax?
\newline
0.3
(
0.0825
f
)
+
0.65
(
0.0825
m
)
0.3(0.0825 f)+0.65(0.0825 m)
0.3
(
0.0825
f
)
+
0.65
(
0.0825
m
)
\newline
(
0.3
+
0.65
)
+
0.0825
(
0.3
+
0.65
)
(0.3+0.65)+0.0825(0.3+0.65)
(
0.3
+
0.65
)
+
0.0825
(
0.3
+
0.65
)
\newline
(
0.3
f
+
0.65
m
)
+
0.0825
(
0.3
f
+
0.65
m
)
(0.3 f+0.65 m)+0.0825(0.3 f+0.65 m)
(
0.3
f
+
0.65
m
)
+
0.0825
(
0.3
f
+
0.65
m
)
\newline
1.0825
(
f
+
m
)
1.0825(f+m)
1.0825
(
f
+
m
)
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The average amount of water in
4
4
4
similar tanks is
k
k
k
litres.
\newline
(a) Express the total amount of water in the tanks in terms of
k
k
k
.
\newline
(b) If
k
=
15
k=15
k
=
15
, find the total amount of water in the
4
4
4
tanks.
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A certain company's main source of income is selling socks. The company's annual profit (in millions of dollars) as a function of the price of a pair of socks (in dollars) is modeled by:
P
(
x
)
=
−
3
(
x
−
5
)
2
+
12
P(x)=-3(x-5)^2+12
P
(
x
)
=
−
3
(
x
−
5
)
2
+
12
What sock price should the company set to earn a maximum profit?
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Uche pumps gasoline at a rate of
18
L
min
\frac{18\,\text{L}}{\text{min}}
min
18
L
. What is Uche's pumping rate in
mL
s
\frac{\text{mL}}{\text{s}}
s
mL
?
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A sheet of glass has a density of
2.5
g
cm
3
\frac{2.5\,\text{g}}{\text{cm}^3}
cm
3
2.5
g
. What is the density of the glass in
kg
m
3
\frac{\text{kg}}{\text{m}^3}
m
3
kg
?
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A certain company's main source of income is selling socks. The company's annual profit (in millions of dollars) as a function of the price of a pair of socks (in dollars) is modeled by:
p
(
x
)
=
−
3
(
x
−
5
)
2
+
12
p(x)=-3(x-5)^2+12
p
(
x
)
=
−
3
(
x
−
5
)
2
+
12
\newline
What is the maximum profit that the company can earn?
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A town has a population of
5.56
×
1
0
4
5.56 \times 10^{4}
5.56
×
1
0
4
and grows at a rate of
4
%
4 \%
4%
every year. Which equation represents the town's population after
7
7
7
years?
\newline
P
=
(
5.56
×
1
0
4
)
(
1
+
0.04
)
7
P=\left(5.56 \times 10^{4}\right)(1+0.04)^{7}
P
=
(
5.56
×
1
0
4
)
(
1
+
0.04
)
7
\newline
P
=
(
5.56
×
1
0
4
)
(
1
+
0.04
)
(
1
+
0.04
)
(
1
+
0.04
)
(
1
+
0.04
)
P=\left(5.56 \times 10^{4}\right)(1+0.04)(1+0.04)(1+0.04)(1+0.04)
P
=
(
5.56
×
1
0
4
)
(
1
+
0.04
)
(
1
+
0.04
)
(
1
+
0.04
)
(
1
+
0.04
)
\newline
P
=
(
5.56
×
1
0
4
)
(
0.04
)
7
P=\left(5.56 \times 10^{4}\right)(0.04)^{7}
P
=
(
5.56
×
1
0
4
)
(
0.04
)
7
\newline
P
=
(
5.56
×
1
0
4
)
(
1
−
0.04
)
7
P=\left(5.56 \times 10^{4}\right)(1-0.04)^{7}
P
=
(
5.56
×
1
0
4
)
(
1
−
0.04
)
7
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T
=
277
+
4
s
T=277+4s
T
=
277
+
4
s
\newline
The temperature in kelvin,
T
T
T
, of a resistor in a circuit
s
s
s
seconds after connecting a battery is given by the equation. For how many seconds does the battery need to be connected for the temperature to increase by
1
1
1
kelvin?
\newline
Choose
1
1
1
answer:
\newline
(A)
0.25
0.25
0.25
\newline
(B)
4
4
4
\newline
(C)
69
69
69
\newline
(D)
277
277
277
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Consumer surplus is the amount of money that consumers save because they are able to buy a product at a price lower than the highest price they would be willing to pay.
\newline
The consumer surplus for a certain product increases at a rate of
900
(
x
+
13
)
−
35
\frac{900}{(x+13)}-35
(
x
+
13
)
900
−
35
dollars per thousand units of the product sold (where
x
x
x
is the number of thousands of units sold).
\newline
By how many dollars does the surplus increase between
x
=
7
x=7
x
=
7
and
x
=
12
x=12
x
=
12
?
\newline
Choose
1
1
1
answer:
\newline
(A)
900
ln
(
0.8
)
−
35
900 \ln (0.8)-35
900
ln
(
0.8
)
−
35
\newline
(B)
900
ln
(
1.25
)
−
35
900 \ln (1.25)-35
900
ln
(
1.25
)
−
35
\newline
(C)
900
ln
(
0.8
)
−
175
900 \ln (0.8)-175
900
ln
(
0.8
)
−
175
\newline
(D)
900
ln
(
1.25
)
−
175
900 \ln (1.25)-175
900
ln
(
1.25
)
−
175
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