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Math Problems
Grade 8
Write and solve direct variation equations
Let
x
x
x
and
y
y
y
be functions of
t
t
t
with
y
=
4
3
π
x
3
y = \frac{4}{3} \pi x^3
y
=
3
4
π
x
3
. If
d
x
d
t
=
1
16
\frac{dx}{dt} = \frac{1}{16}
d
t
d
x
=
16
1
, what is
d
y
d
t
\frac{dy}{dt}
d
t
d
y
when
x
=
6
x = 6
x
=
6
?
\newline
Write an exact, simplified answer.
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If
A
=
{
a
,
b
,
c
,
d
,
e
}
A= \{a, b, c, d, e\}
A
=
{
a
,
b
,
c
,
d
,
e
}
and
B
=
{
c
,
d
,
f
,
g
}
B= \{c, d, f, g\}
B
=
{
c
,
d
,
f
,
g
}
, what is the set difference
B
/
A
B/A
B
/
A
?
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Given
k
(
x
)
=
{
2
x
2
−
4
k(x) = \{2x^2 - 4
k
(
x
)
=
{
2
x
2
−
4
for
x
<
2
x < 2
x
<
2
and
−
3
x
+
10
-3x + 10
−
3
x
+
10
for
x
x
x
is equal to or less than
2
2
2
. Is
k
(
x
)
k(x)
k
(
x
)
continuous at
x
=
2
x=2
x
=
2
? Explain. If
j
(
x
)
=
k
(
x
+
1
)
j(x)=k(x+1)
j
(
x
)
=
k
(
x
+
1
)
, what is the function for
j
(
x
)
j(x)
j
(
x
)
?
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Given
k
(
x
)
=
{
2
x
2
−
4
}
k(x) = \{2x^2 - 4\}
k
(
x
)
=
{
2
x
2
−
4
}
for
x
<
2
x < 2
x
<
2
and
−
3
x
+
10
-3x + 10
−
3
x
+
10
for
x
x
x
is equal to or less than
2
2
2
. Is
k
(
x
)
k(x)
k
(
x
)
continuous at
x
=
2
x=2
x
=
2
? Explain. If
j
(
x
)
=
k
(
x
+
1
+
)
j(x)=k(x+1+)
j
(
x
)
=
k
(
x
+
1
+
)
, what is the function for
j
(
x
)
j(x)
j
(
x
)
?
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Given
k
(
x
)
=
{
2
x
2
−
4
}
k(x) = \{2x^2 - 4\}
k
(
x
)
=
{
2
x
2
−
4
}
for
x
<
2
x < 2
x
<
2
and
−
3
x
+
10
-3x + 10
−
3
x
+
10
for
x
x
x
is equal to or less than
2
2
2
a. Is
k
(
x
)
k(x)
k
(
x
)
continuous at
x
=
2
x=2
x
=
2
? Explain. If
j
(
x
)
=
k
(
x
+
1
+
)
j(x)=k(x+1+)
j
(
x
)
=
k
(
x
+
1
+
)
, what is the function for
j
(
x
)
j(x)
j
(
x
)
?
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8
8
8
−
92
-92
−
92
Given
k
(
x
)
=
{
2
x
2
−
4
for
x
<
2
−
3
x
+
10
for
x
≥
2
k(x)=\left\{\begin{array}{c}2 x^{2}-4 \text { for } x<2 \\ -3 x+10 \text { for } x \geq 2\end{array}\right.
k
(
x
)
=
{
2
x
2
−
4
for
x
<
2
−
3
x
+
10
for
x
≥
2
\newline
a. Is
k
(
x
)
k(x)
k
(
x
)
continuous at
x
=
2
x=2
x
=
2
? Explain.
\newline
b. If
j
(
x
)
=
k
(
x
+
1
)
j(x)=k(x+1)
j
(
x
)
=
k
(
x
+
1
)
, what is the function for
j
(
x
)
j(x)
j
(
x
)
?
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2
2
2
. Suppose
143.0
g
143.0 \mathrm{~g}
143.0
g
aluminum sulfide reacts with
283.0
g
283.0 \mathrm{~g}
283.0
g
of water. What mass of the excess reactant remains? Equation (Balance First):
A
l
2
S
3
+
H
2
O
→
A
l
(
O
H
)
3
\mathrm{Al}_{2} \mathrm{~S}_{3}+\mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{Al}(\mathrm{OH})_{3}
Al
2
S
3
+
H
2
O
→
Al
(
OH
)
3
+
H
2
S
+\mathrm{H}_{2} \mathrm{~S}
+
H
2
S
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Solve the surface area formula
A
=
6
s
2
A=6s^{2}
A
=
6
s
2
for
s
s
s
. Then find the value of
s
s
s
when
A
=
24
A=24
A
=
24
.
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Dashboard
\newline
Similarity
\newline
Similar polygons
\newline
The quadrilaterals.
\newline
J
K
L
M
JKLM
J
K
L
M
and
\newline
P
Q
R
S
PQRS
PQRS
are similar. Find the length
\newline
x
x
x
of
\newline
P
Q
‾
\overline{PQ}
PQ
.
\newline
x
=
□
x=\square
x
=
□
\newline
Explanation
\newline
Check
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To solve
x
−
12
=
95
x-12=95
x
−
12
=
95
you would...
\newline
:ubtract
95
95
95
from both sides
\newline
add
95
95
95
to both sides of the
\newline
add
12
12
12
to both sides of the
\newline
subtract
12
12
12
from both sides of the equation. equation. equation. of the equation.
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If
y
y
y
varies directly with
x
x
x
and
y
=
−
26
y = -26
y
=
−
26
when
x
=
26
x = 26
x
=
26
, find
y
y
y
when
x
=
21
x = 21
x
=
21
. Write and solve a direct variation equation to find the answer.
\newline
y = ____
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A direct variation includes the points
(
6
,
90
)
(6,90)
(
6
,
90
)
and
(
2
,
n
)
(2,n)
(
2
,
n
)
. Find
n
n
n
. Write and solve a direct variation equation to find the answer.
\newline
n = ____
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A direct variation includes the points
(
−
20
,
−
40
)
(-20,-40)
(
−
20
,
−
40
)
and
(
7
,
n
)
(7,n)
(
7
,
n
)
. Find
n
n
n
. Write and solve a direct variation equation to find the answer.
\newline
n
=
_
_
_
n = \_\_\_
n
=
___
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y
=
1
3
x
+
5
y = \frac{1}{3}x+5
y
=
3
1
x
+
5
\newline
y
=
2
x
y= 2x
y
=
2
x
\newline
Consider the given system of equations. If
(
x
,
y
)
(x,y)
(
x
,
y
)
is the solution to the system, then what is the value of
\newline
y
+
x
y+x
y
+
x
?
\newline
◻
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Let
R
R
R
be the region enclosed by the curves
y
=
x
y=\sqrt{x}
y
=
x
and
\newline
y
=
x
3
.
y=\frac{x}{3} \text {. }
y
=
3
x
.
\newline
A solid is generated by rotating
R
R
R
about the line
x
=
−
1
x=-1
x
=
−
1
.
\newline
What is the volume of the solid?
\newline
Give an exact answer in terms of
π
\pi
π
.
\newline
□
\square
□
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3
3
3
. a. Write an equation to represent the nanget.
\newline
b. Explain how to reason with the hanger to find the value of
x
x
x
.
\newline
\$. Explain how to reason with the equation to find the value of \( x \).\(\newline\)\(4\). Andre says that \( x \) is \(7\) because he can move the two \(1\) s with the \( x \) to the other side.\(\newline\)Do you agree with Andre? Explain your reasoning.
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8
8
8
) What is the value of the expression
x
y
x y
x
y
when
x
=
9
x=9
x
=
9
and
y
=
3
y=3
y
=
3
?
\newline
A
6
6
6
\newline
C
27
27
27
\newline
B
12
12
12
\newline
D
93
93
93
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1
1
1
\newline
The table below shows some corresponding values of
x
x
x
and
(
y
−
3
)
2
(y-3)^{2}
(
y
−
3
)
2
where
y
y
y
is positive.
\newline
\begin{tabular}{|c|c|c|c|}
\newline
\hline
x
x
x
&
2
2
2
&
8
8
8
&
32
32
32
\\
\newline
\hline
(
y
−
3
)
2
(y-3)^{2}
(
y
−
3
)
2
&
4
4
4
&
16
16
16
&
64
64
64
\\
\newline
\hline
\newline
\end{tabular}
\newline
x
x
x
\newline
(i) Explain with evidence whether
x
x
x
is directly proportional to
(
y
−
3
)
2
(y-3)^{2}
(
y
−
3
)
2
.
\newline
(ii) Express
x
x
x
in terms of
y
y
y
.
\newline
(iii) Find the value of
x
x
x
when
(
y
−
3
)
2
(y-3)^{2}
(
y
−
3
)
2
1
1
1
.
\newline
(iv) Find the value of
y
y
y
when
(
y
−
3
)
2
(y-3)^{2}
(
y
−
3
)
2
3
3
3
.
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32
32
32
. For functions
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
)
,
g
(
x
)
=
f
(
x
+
3
)
−
12
g(x), g(x)=f(x+3)-12
g
(
x
)
,
g
(
x
)
=
f
(
x
+
3
)
−
12
. If
f
(
4
)
=
112
f(4)=112
f
(
4
)
=
112
, what is the value of
g
(
1
)
g(1)
g
(
1
)
?
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If
f
(
1
)
=
9
f(1)=9
f
(
1
)
=
9
and
f
(
n
)
=
−
4
f
(
n
−
1
)
f(n)=-4 f(n-1)
f
(
n
)
=
−
4
f
(
n
−
1
)
then find the value of
f
(
4
)
f(4)
f
(
4
)
.
\newline
Answer:
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Solve the following equation for
x
x
x
. Express your answer in the simplest form.
\newline
4
(
7
x
+
5
)
=
28
x
+
21
4(7 x+5)=28 x+21
4
(
7
x
+
5
)
=
28
x
+
21
\newline
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Solve the following equation for
x
x
x
. Express your answer in the simplest form.
\newline
−
5
(
−
4
x
+
3
)
=
16
x
−
16
-5(-4 x+3)=16 x-16
−
5
(
−
4
x
+
3
)
=
16
x
−
16
\newline
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12
t
=
4
v
−
3
12t = 4v - 3
12
t
=
4
v
−
3
\newline
−
6
t
=
4
v
+
6
-6t = 4v + 6
−
6
t
=
4
v
+
6
\newline
If
(
t
,
v
)
(t, v)
(
t
,
v
)
is the solution to the system of equations, what is the value of
t
−
v
t - v
t
−
v
?
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Use the given information to find the unknown value:
y
y
y
varies directly as the cube of
x
x
x
. When
x
=
2
x=2
x
=
2
, then
y
=
16
y=16
y
=
16
. Find
y
y
y
when
x
=
3
x=3
x
=
3
.
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If
y
y
y
varies directly with
x
x
x
and
y
=
15
y = 15
y
=
15
when
x
=
5
x = 5
x
=
5
, find
y
y
y
when
x
=
1
x = 1
x
=
1
. Write and solve a direct variation equation to find the answer.
\newline
y
y
y
= ____
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If
y
y
y
varies directly with
x
x
x
and
y
=
20
y = 20
y
=
20
when
x
=
10
x = 10
x
=
10
, find
y
y
y
when
x
=
4
x = 4
x
=
4
. Write and solve a direct variation equation to find the answer.
\newline
y
=
‾
y = \underline{\hspace{2em}}
y
=
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If
y
y
y
varies directly with
x
x
x
and
y
=
14
y = 14
y
=
14
when
x
=
2
x = 2
x
=
2
, find
y
y
y
when
x
=
1
x = 1
x
=
1
. Write and solve a direct variation equation to find the answer.
\newline
y
y
y
= ____
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If
y
y
y
varies directly with
x
x
x
and
y
=
8
y = 8
y
=
8
when
x
=
4
x = 4
x
=
4
, find
y
y
y
when
x
=
3
x = 3
x
=
3
. Write and solve a direct variation equation to find the answer.
\newline
y
y
y
= ___
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If
y
y
y
varies directly with
x
x
x
and
y
=
12
y = 12
y
=
12
when
x
=
4
x = 4
x
=
4
, find
y
y
y
when
x
=
1
x = 1
x
=
1
. Write and solve a direct variation equation to find the answer.
\newline
y
y
y
= ____
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A direct variation includes the points
(
4
,
16
)
(4,16)
(
4
,
16
)
and
(
1
,
n
)
(1,n)
(
1
,
n
)
. Find
n
n
n
. Write and solve a direct variation equation to find the answer.
\newline
n
=
_
_
_
n = \_\_\_
n
=
___
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A direct variation includes the points
(
3
,
15
)
(3,15)
(
3
,
15
)
and
(
1
,
n
)
(1,n)
(
1
,
n
)
. Find
n
n
n
. Write and solve a direct variation equation to find the answer.
\newline
n
=
_
_
_
n = \_\_\_
n
=
___
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If
ℓ
=
2
(
w
+
1
)
\ell=2(w+1)
ℓ
=
2
(
w
+
1
)
, which of the following correctly gives
w
w
w
in terms of
ℓ
\ell
ℓ
?
\newline
Choose
1
1
1
answer:
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egin{
−
20
-20
−
20
x+
12
12
12
y=
24
24
24
} and egin{
−
5
-5
−
5
x+
2
2
2
y= rac{
6
6
6
}{}} consider the system of equations. how many egin{(x,y)} solutions does this system have?
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If
x
x
x
and
y
y
y
are in direct proportion and
y
y
y
is
36
36
36
when
x
x
x
is
9
9
9
, find
y
y
y
when
x
x
x
is
8
8
8
.
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If
x
x
x
and
y
y
y
are in direct proportion and
y
y
y
is
28
28
28
when
x
x
x
is
4
4
4
, find
y
y
y
when
x
x
x
is
3
3
3
.
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Find the mean of the
32
32
32
numbers, such that if the mean of
10
10
10
of them is
15
15
15
and the mean of
20
20
20
of them is
11
11
11
. The last two numbers are
10
10
10
.
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what is the value of
x
2
y
4
\dfrac{x^2}{y^4}
y
4
x
2
when
x
=
8
x=8
x
=
8
and
y
=
2
y=2
y
=
2
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If
a
1
=
5
,
a
2
=
1
a_{1}=5, a_{2}=1
a
1
=
5
,
a
2
=
1
and
a
n
=
3
a
n
−
1
−
2
a
n
−
2
a_{n}=3 a_{n-1}-2 a_{n-2}
a
n
=
3
a
n
−
1
−
2
a
n
−
2
then find the value of
a
6
a_{6}
a
6
.
\newline
Answer:
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If
a
1
=
4
,
a
2
=
1
a_{1}=4, a_{2}=1
a
1
=
4
,
a
2
=
1
and
a
n
=
2
a
n
−
1
+
3
a
n
−
2
a_{n}=2 a_{n-1}+3 a_{n-2}
a
n
=
2
a
n
−
1
+
3
a
n
−
2
then find the value of
a
4
a_{4}
a
4
.
\newline
Answer:
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If
a
1
=
4
,
a
2
=
5
a_{1}=4, a_{2}=5
a
1
=
4
,
a
2
=
5
and
a
n
=
2
a
n
−
1
+
a
n
−
2
a_{n}=2 a_{n-1}+a_{n-2}
a
n
=
2
a
n
−
1
+
a
n
−
2
then find the value of
a
4
a_{4}
a
4
.
\newline
Answer:
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If
a
1
=
0
,
a
2
=
3
a_{1}=0, a_{2}=3
a
1
=
0
,
a
2
=
3
and
a
n
=
3
a
n
−
1
−
a
n
−
2
a_{n}=3 a_{n-1}-a_{n-2}
a
n
=
3
a
n
−
1
−
a
n
−
2
then find the value of
a
4
a_{4}
a
4
.
\newline
Answer:
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If
a
1
=
2
,
a
2
=
0
a_{1}=2, a_{2}=0
a
1
=
2
,
a
2
=
0
and
a
n
=
2
a
n
−
1
−
3
a
n
−
2
a_{n}=2 a_{n-1}-3 a_{n-2}
a
n
=
2
a
n
−
1
−
3
a
n
−
2
then find the value of
a
5
a_{5}
a
5
.
\newline
Answer:
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If
a
1
=
5
,
a
2
=
0
a_{1}=5, a_{2}=0
a
1
=
5
,
a
2
=
0
and
a
n
=
a
n
−
1
+
3
a
n
−
2
a_{n}=a_{n-1}+3 a_{n-2}
a
n
=
a
n
−
1
+
3
a
n
−
2
then find the value of
a
5
a_{5}
a
5
.
\newline
Answer:
Get tutor help
If
a
1
=
0
,
a
2
=
3
a_{1}=0, a_{2}=3
a
1
=
0
,
a
2
=
3
and
a
n
=
2
a
n
−
1
−
a
n
−
2
a_{n}=2 a_{n-1}-a_{n-2}
a
n
=
2
a
n
−
1
−
a
n
−
2
then find the value of
a
4
a_{4}
a
4
.
\newline
Answer:
Get tutor help
If
a
1
=
5
,
a
2
=
2
a_{1}=5, a_{2}=2
a
1
=
5
,
a
2
=
2
and
a
n
=
3
a
n
−
1
+
2
a
n
−
2
a_{n}=3 a_{n-1}+2 a_{n-2}
a
n
=
3
a
n
−
1
+
2
a
n
−
2
then find the value of
a
4
a_{4}
a
4
.
\newline
Answer:
Get tutor help
If
a
1
=
2
,
a
2
=
5
a_{1}=2, a_{2}=5
a
1
=
2
,
a
2
=
5
and
a
n
=
2
a
n
−
1
−
a
n
−
2
a_{n}=2 a_{n-1}-a_{n-2}
a
n
=
2
a
n
−
1
−
a
n
−
2
then find the value of
a
5
a_{5}
a
5
.
\newline
Answer:
Get tutor help
If
a
1
=
4
,
a
2
=
2
a_{1}=4, a_{2}=2
a
1
=
4
,
a
2
=
2
and
a
n
=
2
a
n
−
1
+
a
n
−
2
a_{n}=2 a_{n-1}+a_{n-2}
a
n
=
2
a
n
−
1
+
a
n
−
2
then find the value of
a
6
a_{6}
a
6
.
\newline
Answer:
Get tutor help
If
a
1
=
2
,
a
2
=
1
a_{1}=2, a_{2}=1
a
1
=
2
,
a
2
=
1
and
a
n
=
2
a
n
−
1
+
a
n
−
2
a_{n}=2 a_{n-1}+a_{n-2}
a
n
=
2
a
n
−
1
+
a
n
−
2
then find the value of
a
5
a_{5}
a
5
.
\newline
Answer:
Get tutor help
If
a
1
=
3
,
a
2
=
1
a_{1}=3, a_{2}=1
a
1
=
3
,
a
2
=
1
and
a
n
=
3
a
n
−
1
−
2
a
n
−
2
a_{n}=3 a_{n-1}-2 a_{n-2}
a
n
=
3
a
n
−
1
−
2
a
n
−
2
then find the value of
a
5
a_{5}
a
5
.
\newline
Answer:
Get tutor help
If
f
(
1
)
=
1
,
f
(
2
)
=
0
f(1)=1, f(2)=0
f
(
1
)
=
1
,
f
(
2
)
=
0
and
f
(
n
)
=
f
(
n
−
1
)
−
3
f
(
n
−
2
)
f(n)=f(n-1)-3 f(n-2)
f
(
n
)
=
f
(
n
−
1
)
−
3
f
(
n
−
2
)
then find the value of
f
(
6
)
f(6)
f
(
6
)
.
\newline
Answer:
Get tutor help
1
2
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