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Math Problems
Grade 6
Find a value using two-variable equations
f
(
x
)
=
10
(
1.25
)
x
f(x)=10(1.25)^x
f
(
x
)
=
10
(
1.25
)
x
\newline
The function models
f
f
f
, the price of a rare trading card in dollars
x
x
x
years after its initial release in
1993
1993
1993
. Based on the model, what is the price of the trading card
20
20
20
years after its initial release?
\newline
Choose
1
1
1
answer:
\newline
(A)
$
\$
$
15
15
15
.
63
63
63
\newline
(B)
$
\$
$
16
16
16
.
39
39
39
\newline
(C)
$
\$
$
93
93
93
.
13
13
13
\newline
(D)
$
\$
$
867
867
867
.
36
36
36
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The rate of change
d
P
d
t
\frac{d P}{d t}
d
t
d
P
of the number of students who heard a rumor is modeled by a logistic differential equation. The maximum capacity of the school is
858
858
858
students. At
6
A
M
6 \mathrm{AM}
6
AM
, the number of students who heard the rumor is
220
220
220
and is increasing at a rate of
37
37
37
students per hour. Write a differential equation to describe the situation.
\newline
d
P
d
t
=
□
\frac{d P}{d t}=\square
d
t
d
P
=
□
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Which of the following are true statements about the expression
\newline
6
+
(
−
6
)
6+(-6)
6
+
(
−
6
)
?
\newline
Choose
2
2
2
answers:
\newline
A The expression describes the number that is
6
6
6
to the left of
6
6
6
on the number line.
\newline
B The expression describes the number that is
6
6
6
to the right of
6
6
6
on the number line.
\newline
C The expression equals
0
0
0
.
\newline
D The expression equals
12
12
12
.
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Which of the following are true statements about the expression
\newline
−
9
+
9
-9+9
−
9
+
9
?
\newline
Choose
2
2
2
answers:
\newline
A The expression describes the number that is
9
9
9
to the left of
−
9
-9
−
9
on the number line.
\newline
B The expression describes the number that is
9
9
9
to the right of
−
9
-9
−
9
on the number line.
\newline
C The expression equals
−
18
-18
−
18
.
\newline
D The expression equals
0
0
0
.
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Which of the following are true statements about the expression
\newline
−
8
+
8
-8+8
−
8
+
8
?
\newline
Choose
2
2
2
answers:
\newline
A The expression describes the number that is
8
8
8
to the left of
−
8
-8
−
8
on the number line.
\newline
B The expression describes the number that is
8
8
8
to the right of
−
8
-8
−
8
on the number line.
\newline
C The expression equals
−
16
-16
−
16
.
\newline
D The expression equals
0
0
0
.
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A single processor takes
20
20
20
milliseconds (
m
s
\mathrm{ms}
ms
) to prepare data entries and
0.1
n
m
s
0.1 n \mathrm{~ms}
0.1
n
ms
to copy the entries, where
n
n
n
is the number of entries. A multiprocessor takes
70
m
s
70 \mathrm{~ms}
70
ms
to prepare and copy one data entry, and whenever the number of entries is doubled the amount of time to prepare and copy them increases by
5
m
s
5 \mathrm{~ms}
5
ms
. Given
120
m
s
120 \mathrm{~ms}
120
ms
to prepare and copy data entries, which processor type can prepare and copy more entries and how many more entries can it prepare and copy?
\newline
Choose
1
1
1
answer:
\newline
(A) The single processor can prepare and copy
176
176
176
more entries.
\newline
(B) The single processor can prepare and copy
989
989
989
more entries.
\newline
(C) The multiprocessor can prepare and copy
24
24
24
more entries.
\newline
(D) The multiprocessor can prepare and copy
1
1
1
,
012
012
012
more entries.
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Sarah was given this problem:
\newline
The base
b
(
t
)
b(t)
b
(
t
)
of a triangle is decreasing at a rate of
13
13
13
millimeters per minute and the height
h
(
t
)
h(t)
h
(
t
)
of the triangle is increasing at a rate of
6
6
6
millimeters per minute. At a certain instant
t
0
t_{0}
t
0
, the base is
5
5
5
millimeters and the height is
1
1
1
millimeter. What is the rate of change of the area
A
(
t
)
A(t)
A
(
t
)
of the triangle at that instant (in square millimeters per minute)?
\newline
Which equation should Sarah use to solve the problem?
\newline
Choose
1
1
1
answer:
\newline
(A)
A
(
t
)
=
b
(
t
)
⋅
h
(
t
)
2
A(t)=\frac{b(t) \cdot h(t)}{2}
A
(
t
)
=
2
b
(
t
)
⋅
h
(
t
)
\newline
(B)
A
(
t
)
=
b
(
t
)
⋅
h
(
t
)
A(t)=b(t) \cdot h(t)
A
(
t
)
=
b
(
t
)
⋅
h
(
t
)
\newline
(C)
A
(
t
)
+
b
(
t
)
+
h
(
t
)
=
180
A(t)+b(t)+h(t)=180
A
(
t
)
+
b
(
t
)
+
h
(
t
)
=
180
\newline
(D)
[
A
(
t
)
]
2
=
[
b
(
t
)
]
2
+
[
h
(
t
)
]
2
[A(t)]^{2}=[b(t)]^{2}+[h(t)]^{2}
[
A
(
t
)
]
2
=
[
b
(
t
)
]
2
+
[
h
(
t
)
]
2
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Lúcia was given this problem:
\newline
A
15
15
15
-meter ladder is leaning against a wall. The distance
y
(
t
)
y(t)
y
(
t
)
between the top of the ladder and the ground is decreasing at a rate of
6
6
6
meters per minute. At a certain instant
t
0
t_{0}
t
0
, the bottom of the ladder is a distance
x
(
t
0
)
x\left(t_{0}\right)
x
(
t
0
)
of
12
12
12
meters from the wall. What is the rate of change of the angle
θ
(
t
)
\theta(t)
θ
(
t
)
between the ground and the ladder at that instant?
\newline
Which equation should Lúcia use to solve the problem?
\newline
Choose
1
1
1
answer:
\newline
(A)
sin
[
θ
(
t
)
]
=
y
(
t
)
15
\sin [\theta(t)]=\frac{y(t)}{15}
sin
[
θ
(
t
)]
=
15
y
(
t
)
\newline
(B)
θ
(
t
)
=
x
(
t
)
⋅
y
(
t
)
2
\theta(t)=\frac{x(t) \cdot y(t)}{2}
θ
(
t
)
=
2
x
(
t
)
⋅
y
(
t
)
\newline
(C)
θ
(
t
)
+
x
(
t
)
+
y
(
t
)
=
180
\theta(t)+x(t)+y(t)=180
θ
(
t
)
+
x
(
t
)
+
y
(
t
)
=
180
\newline
(D)
[
θ
(
t
)
]
2
=
[
x
(
t
)
]
2
+
[
y
(
t
)
]
2
[\theta(t)]^{2}=[x(t)]^{2}+[y(t)]^{2}
[
θ
(
t
)
]
2
=
[
x
(
t
)
]
2
+
[
y
(
t
)
]
2
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Calvin and Melvin are two of Santa's elves. The number of toys they build is given by
11
c
+
8
m
11 c+8 m
11
c
+
8
m
where
c
c
c
is the number of candy canes Calvin eats for breakfast and
m
m
m
is the number of candy canes Melvin eats for breakfast.
\newline
How many toys do they build when Calvin eats
5
5
5
candy canes and Melvin eats
3
3
3
candy canes for breakfast?
\newline
toys
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Calvin and Melvin are two of Santa's elves. The number of toys they build is given by
11
c
+
8
m
11 c+8 m
11
c
+
8
m
, where
c
c
c
is the number of candy canes Calvin eats for breakfast and
m
m
m
is the number of candy canes Melvin eats for breakfast. How many toys do they build when Calvin eats
5
5
5
candy canes and Melvin eats
3
3
3
candy canes for breakfast?
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2
w
+
4
2 w+4
2
w
+
4
\newline
Carly is cutting a piece of paper stock based on the width of a photo. The expression above describes the length of the paper stock, in inches, for a photo that is
w
w
w
inches wide. If the photo is
8
8
8
.
5
5
5
inches wide, what is the length of the paper stock in inches?
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This equation shows how the total bill for a meal is related to the number of people sharing the meal.
\newline
b
=
15
n
+
35
b = 15n + 35
b
=
15
n
+
35
\newline
The variable
n
n
n
represents the number of people, and the variable
b
b
b
represents the total bill. What is the total bill for a meal shared by
4
4
4
people?
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The equation shows how the amount of paint needed is related to the area to be painted.
p
=
2
a
+
5
p = 2a + 5
p
=
2
a
+
5
. The variable
a
a
a
represents the area in square meters, and the variable
p
p
p
represents the amount of paint in liters. How much paint is needed for an area of
8
8
8
square meters?
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This equation shows how the total score in a game is related to the number of points scored per level.
s
=
100
l
+
250
s = 100l + 250
s
=
100
l
+
250
. The variable
l
l
l
represents the levels completed, and the variable
s
s
s
represents the total score. What is the total score after completing
3
3
3
levels?
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The equation shows how the cost of buying pencils is related to the number of pencils purchased.
\newline
c
=
0.5
p
+
2
c= 0.5p+2
c
=
0.5
p
+
2
\newline
The variable
p
p
p
represents the number of pencils, and the variable
c
c
c
represents the total cost. How much does it cost to buy
10
10
10
pencils?
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This equation shows how the distance traveled by a car is related to the time spent driving.
d
=
50
t
+
10
d = 50t + 10
d
=
50
t
+
10
. The variable
t
t
t
represents the time in hours, and the variable
d
d
d
represents the distance in miles. How far does the car travel in
2
2
2
hours?
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