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Math Problems
Algebra 2
Write and solve direct variation equations
Let
x
x
x
and
y
y
y
be functions of
t
t
t
with
y
=
x
2
+
x
1
2
y = x^2 + x^{\frac{1}{2}}
y
=
x
2
+
x
2
1
. If
d
x
d
t
=
1
8
\frac{dx}{dt} = \frac{1}{8}
d
t
d
x
=
8
1
, what is
d
y
d
t
\frac{dy}{dt}
d
t
d
y
when
x
=
4
x = 4
x
=
4
?
\newline
Write an exact, simplified answer.
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Let
x
x
x
and
y
y
y
be functions of
t
t
t
with
y
=
x
1
2
y = x^{\frac{1}{2}}
y
=
x
2
1
. If
d
x
d
t
=
4
\frac{dx}{dt} = 4
d
t
d
x
=
4
, what is
d
y
d
t
\frac{dy}{dt}
d
t
d
y
when
x
=
1
x = 1
x
=
1
?
\newline
Write an exact, simplified answer.
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Let
x
x
x
and
y
y
y
be functions of
t
t
t
with
y
=
4
π
x
3
y = 4 \pi x^3
y
=
4
π
x
3
. If
d
x
d
t
=
6
\frac{dx}{dt} = 6
d
t
d
x
=
6
, what is
d
y
d
t
\frac{dy}{dt}
d
t
d
y
when
x
=
1
3
x = \frac{1}{3}
x
=
3
1
?
\newline
Write an exact, simplified answer.
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Let
x
x
x
and
y
y
y
be functions of
t
t
t
with
y
=
16
x
2
+
4
x
+
3
y = 16x^2 + 4x + 3
y
=
16
x
2
+
4
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
=
4
x = 4
x
=
4
?
\newline
Write an exact, simplified answer.
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Let
x
x
x
and
y
y
y
be functions of
t
t
t
with
y
=
4
sin
x
y = 4\sin x
y
=
4
sin
x
. If
d
x
d
t
=
5
\frac{dx}{dt} = 5
d
t
d
x
=
5
, what is
d
y
d
t
\frac{dy}{dt}
d
t
d
y
when
x
=
π
3
x = \frac{\pi}{3}
x
=
3
π
?
\newline
Write an exact, simplified answer.
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Let
x
x
x
and
y
y
y
be functions of
t
t
t
with
y
=
4
e
x
y = 4e^x
y
=
4
e
x
. If
d
x
d
t
=
1
4
\frac{dx}{dt} = \frac{1}{4}
d
t
d
x
=
4
1
, what is
d
y
d
t
\frac{dy}{dt}
d
t
d
y
when
x
=
2
x = 2
x
=
2
?
\newline
Write an exact, simplified answer.
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Let
x
x
x
and
y
y
y
be functions of
t
t
t
with
y
=
2
x
3
+
4
x
2
+
3
y = 2x^3 + 4x^2 + 3
y
=
2
x
3
+
4
x
2
+
3
. If
d
x
d
t
=
1
3
\frac{dx}{dt} = \frac{1}{3}
d
t
d
x
=
3
1
, what is
d
y
d
t
\frac{dy}{dt}
d
t
d
y
when
x
=
4
x = 4
x
=
4
?
\newline
Write an exact, simplified answer.
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Let
x
x
x
and
y
y
y
be functions of
t
t
t
with
y
=
x
1
3
y = x^{\frac{1}{3}}
y
=
x
3
1
. If
d
x
d
t
=
7
\frac{dx}{dt} = 7
d
t
d
x
=
7
, what is
d
y
d
t
\frac{dy}{dt}
d
t
d
y
when
x
=
8
x = 8
x
=
8
?
\newline
Write an exact, simplified answer.
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Let
x
x
x
and
y
y
y
be functions of
t
t
t
with
y
=
4
π
x
2
y = 4 \pi x^2
y
=
4
π
x
2
. If
d
x
d
t
=
−
1
3
\frac{dx}{dt} = -\frac{1}{3}
d
t
d
x
=
−
3
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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Let
x
x
x
and
y
y
y
be functions of
t
t
t
with
y
=
sin
x
y = \sin x
y
=
sin
x
. If
d
x
d
t
=
3
\frac{dx}{dt} = 3
d
t
d
x
=
3
, what is
d
y
d
t
\frac{dy}{dt}
d
t
d
y
when
x
=
π
x = \pi
x
=
π
?
\newline
Write an exact, simplified answer.
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Let
x
x
x
and
y
y
y
be functions of
t
t
t
with
y
=
π
x
2
y = \pi x^2
y
=
π
x
2
. If
d
x
d
t
=
−
1
8
\frac{dx}{dt} = -\frac{1}{8}
d
t
d
x
=
−
8
1
, what is
d
y
d
t
\frac{dy}{dt}
d
t
d
y
when
x
=
16
x = 16
x
=
16
?
\newline
Write an exact, simplified answer.
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Let
x
x
x
and
y
y
y
be functions of
t
t
t
with
y
=
tan
x
y = \tan x
y
=
tan
x
. If
d
x
d
t
=
5
\frac{dx}{dt} = 5
d
t
d
x
=
5
, what is
d
y
d
t
\frac{dy}{dt}
d
t
d
y
when
x
=
π
4
x = \frac{\pi}{4}
x
=
4
π
?
\newline
Write an exact, simplified answer.
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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
8
\frac{dx}{dt} = \frac{1}{8}
d
t
d
x
=
8
1
, what is
d
y
d
t
\frac{dy}{dt}
d
t
d
y
when
x
=
3
x = 3
x
=
3
?
\newline
Write an exact, simplified answer.
Get tutor help
Let
x
x
x
and
y
y
y
be functions of
t
t
t
with
y
=
7
x
y = 7x
y
=
7
x
. If
d
x
d
t
=
2
\frac{dx}{dt} = 2
d
t
d
x
=
2
, what is
d
y
d
t
\frac{dy}{dt}
d
t
d
y
when
x
=
π
2
x = \frac{\pi}{2}
x
=
2
π
?
\newline
Write an exact, simplified answer.
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Let
ζ
\zeta
ζ
be a point selected at random uniformly from the unit interval
[
0
,
1
[0,1
[
0
,
1
Consider the random variable
X
=
e
ζ
X=e^{\zeta}
X
=
e
ζ
.
\newline
(a) Sketch
X
X
X
as a function of
ζ
\zeta
ζ
.
\newline
(b) Find and plot the CDF of
X
X
X
.
\newline
(c) Find the probability of the events
{
X
>
e
}
\{X>\sqrt{e}\}
{
X
>
e
}
and
{
e
0.25
<
X
≤
e
0.75
}
\left\{e^{0.25}<X \leq e^{0.75}\right\}
{
e
0.25
<
X
≤
e
0.75
}
.
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Write an equation that describes the following relationship:
y
y
y
varies directly as the square of
x
x
x
and when
x
=
4
,
y
=
80
x=4, y=80
x
=
4
,
y
=
80
.
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In a direct variation,
y
=
20
y = 20
y
=
20
when
x
=
2
x = 2
x
=
2
. Write a direct variation equation that shows the relationship between
x
x
x
and
y
y
y
.
\newline
Write your answer as an equation with
y
y
y
first, followed by an equals sign.
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In a direct variation,
y
=
10
y = 10
y
=
10
when
x
=
5
x = 5
x
=
5
. Write a direct variation equation that shows the relationship between
x
x
x
and
y
y
y
.
\newline
Write your answer as an equation with
y
y
y
first, followed by an equals sign.
Get tutor help
In a direct variation,
y
=
18
y = 18
y
=
18
when
x
=
2
x = 2
x
=
2
. Write a direct variation equation that shows the relationship between
x
x
x
and
y
y
y
.
\newline
Write your answer as an equation with
y
y
y
first, followed by an equals sign.
Get tutor help
In a direct variation,
y
=
4
y = 4
y
=
4
when
x
=
2
x = 2
x
=
2
. Write a direct variation equation that shows the relationship between
x
x
x
and
y
y
y
.
\newline
Write your answer as an equation with
y
y
y
first, followed by an equals sign.
Get tutor help
Solve for
b
b
b
in the proportion.
\newline
10
8
=
b
12
b
=
□
\begin{array}{l}\frac{10}{8}=\frac{b}{12} \\b=\square\end{array}
8
10
=
12
b
b
=
□
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Suppose that
z
z
z
varies jointly with the cube of
x
x
x
and the square of
y
y
y
. Find the constant of proportionality
k
k
k
if
z
=
3600
z = 3600
z
=
3600
when
y
=
3
y = 3
y
=
3
and
x
=
5
x =5
x
=
5
. Using the
k
k
k
from above write the variation equation in terms of
x
x
x
and
y
y
y
. Using the
k
k
k
from above find
z
z
z
given that
x
=
x =
x
=
12
12
12
and
x
=
x=
x
=
13
13
13
.
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Suppose that
z
z
z
varies jointly with the cube of
x
x
x
and the square of
y
y
y
. Find the constant of proportionality
k
k
k
if
z
=
3600
z = 3600
z
=
3600
when
y
=
3
y = 3
y
=
3
and
x
=
5
x =5
x
=
5
. Using the
k
k
k
from above write the variation equation in terms of
x
x
x
and
y
y
y
. Using the
k
k
k
from above find
z
z
z
given that
x
=
x=
x
=
12
12
12
and
x
=
x=
x
=
13
13
13
.
Get tutor help
Suppose that
z
z
z
varies jointly with the cube of
x
x
x
and the square of
y
y
y
. Find the constant of proportionality
k
k
k
if
z
=
3600
z = 3600
z
=
3600
when
y
=
3
y = 3
y
=
3
and
x
=
5
x =5
x
=
5
.
Get tutor help
Suppose that
z
z
z
varies jointly with the cube of
x
x
x
and the square of
y
y
y
. Find the constant of proportionality
k
k
k
if
z
=
3600
z = 3600
z
=
3600
when
y
=
3
y = 3
y
=
3
and
x
=
5
x =5
x
=
5
. Using the
k
k
k
from above write the variation equation in terms of
x
x
x
and
y
y
y
.
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Square VWXY is dilated by a scale factor of
6
6
6
to form square
V
′
W
′
X
′
Y
′
V^{\prime} W^{\prime} X^{\prime} Y^{\prime}
V
′
W
′
X
′
Y
′
. Side W'X' measures
12
12
12
. What is the measure of side WX?
\newline
Answer:
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Given the substitutions
ln
2
=
a
,
ln
3
=
b
\ln 2=a, \ln 3=b
ln
2
=
a
,
ln
3
=
b
, and
ln
5
=
c
\ln 5=c
ln
5
=
c
, find the value of
ln
(
e
3
)
\ln \left(\frac{e}{3}\right)
ln
(
3
e
)
in terms of
a
,
b
a, b
a
,
b
, and
c
c
c
.
\newline
Answer:
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In a direct variation,
y
=
15
y=15
y
=
15
when
x
=
5
x=5
x
=
5
. Write a direct variation equation that shows the relationship between
x
x
x
and
y
y
y
. Write your answer as an equation with
y
y
y
first, followed by an equals sign.
Get tutor help
