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Math Problems
Precalculus
Find trigonometric ratios using a Pythagorean or reciprocal identity
Let
x
x
x
and
y
y
y
be functions of
t
t
t
with
y
=
15
cos
x
y = 15\cos x
y
=
15
cos
x
. If
d
x
d
t
=
−
1
9
\frac{dx}{dt} = -\frac{1}{9}
d
t
d
x
=
−
9
1
, what is
d
y
d
t
\frac{dy}{dt}
d
t
d
y
when
x
=
π
6
x = \frac{\pi}{6}
x
=
6
π
?
\newline
Write an exact, simplified answer.
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The planet Neptune orbits the Sun. Its orbital radius is
30
30
30
.
1
1
1
astronomical units (AU).
\newline
Assuming Neptune's orbit is circular, what is the distance it travels in a single orbit around the Sun?
\newline
Give your answer in terms of
π
\pi
π
.
\newline
□
\square
□
AU
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For the rotation
−
15
2
∘
-152^{\circ}
−
15
2
∘
, find the coterminal angle from
0
∘
≤
θ
<
36
0
∘
0^{\circ} \leq \theta<360^{\circ}
0
∘
≤
θ
<
36
0
∘
, the quadrant, and the reference angle.
\newline
The coterminal angle is
□
∘
\square^{\circ}
□
∘
, which lies in Quadrant
□
\square
□
, with a reference angle of
□
∘
\square^{\circ}
□
∘
.
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For the rotation
−
105
3
∘
-1053^{\circ}
−
105
3
∘
, find the coterminal angle from
0
∘
≤
θ
<
36
0
∘
0^{\circ} \leq \theta<360^{\circ}
0
∘
≤
θ
<
36
0
∘
, the quadrant, and the reference angle.
\newline
The coterminal angle is
□
∘
\square^{\circ}
□
∘
, which lies in Quadrant
□
\square
□
, with a reference angle of
□
∘
\square^{\circ}
□
∘
.
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
4
,
7
)
(4,7)
(
4
,
7
)
and a terminal point at
(
−
1
,
7
)
(-1,7)
(
−
1
,
7
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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simplify this
r
=
C
p
C
v
=
C
v
+
R
C
v
=
R
r
+
1
+
R
R
/
r
+
1
r = \frac{C_p}{C_v} = \frac{C_v + R}{C_v} = \frac{R}{r} + 1 + \frac{R}{R/r} + 1
r
=
C
v
C
p
=
C
v
C
v
+
R
=
r
R
+
1
+
R
/
r
R
+
1
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The angle
θ
1
\theta_{1}
θ
1
is located in Quadrant III, and
sin
(
θ
1
)
=
−
12
13
\sin \left(\theta_{1}\right)=-\frac{12}{13}
sin
(
θ
1
)
=
−
13
12
.
\newline
What is the value of
cos
(
θ
1
)
\cos \left(\theta_{1}\right)
cos
(
θ
1
)
?
\newline
Express your answer exactly.
\newline
cos
(
θ
1
)
=
\cos \left(\theta_{1}\right)=
cos
(
θ
1
)
=
\newline
□
\square
□
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What is the exact value of
tan
(
19
π
12
)
\tan(\frac{19\pi}{12})
tan
(
12
19
π
)
?
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Given that
cos
A
=
2
3
\cos A=\frac{\sqrt{2}}{3}
cos
A
=
3
2
and
cos
B
=
30
6
\cos B=\frac{\sqrt{30}}{6}
cos
B
=
6
30
, and that angles
A
A
A
and
B
B
B
are both in Quadrant I, find the exact value of
sin
(
A
−
B
)
\sin (A-B)
sin
(
A
−
B
)
, in simplest radical form.
\newline
Answer:
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Given that
sin
x
=
24
6
\sin x=\frac{\sqrt{24}}{6}
sin
x
=
6
24
and
sin
y
=
5
3
\sin y=\frac{\sqrt{5}}{3}
sin
y
=
3
5
, and that angles
x
x
x
and
y
y
y
are both in Quadrant I, find the exact value of
sin
(
x
+
y
)
\sin (x+y)
sin
(
x
+
y
)
, in simplest radical form.
\newline
Answer:
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Given that
tan
A
=
1
2
\tan A=\frac{1}{2}
tan
A
=
2
1
and
sin
B
=
6
85
\sin B=\frac{6}{\sqrt{85}}
sin
B
=
85
6
, and that angles
A
A
A
and
B
B
B
are both in Quadrant I, find the exact value of
cos
(
A
+
B
)
\cos (A+B)
cos
(
A
+
B
)
, in simplest radical form.
\newline
Answer:
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Given that
cos
x
=
3
10
\cos x=\frac{3}{\sqrt{10}}
cos
x
=
10
3
and
tan
y
=
3
2
\tan y=\frac{3}{2}
tan
y
=
2
3
, and that angles
x
x
x
and
y
y
y
are both in Quadrant I, find the exact value of
sin
(
x
−
y
)
\sin (x-y)
sin
(
x
−
y
)
, in simplest radical form.
\newline
Answer:
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Given that
tan
x
=
3
\tan x=\sqrt{3}
tan
x
=
3
and
cos
y
=
2
2
\cos y=\frac{\sqrt{2}}{2}
cos
y
=
2
2
, and that angles
x
x
x
and
y
y
y
are both in Quadrant I, find the exact value of
cos
(
x
+
y
)
\cos (x+y)
cos
(
x
+
y
)
, in simplest radical form.
\newline
Answer:
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Given that
cos
A
=
20
5
\cos A=\frac{\sqrt{20}}{5}
cos
A
=
5
20
and
cos
B
=
6
3
\cos B=\frac{\sqrt{6}}{3}
cos
B
=
3
6
, and that angles
A
A
A
and
B
B
B
are both in Quadrant I, find the exact value of
sin
(
A
−
B
)
\sin (A-B)
sin
(
A
−
B
)
, in simplest radical form.
\newline
Answer:
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Given the reference angle of
2
π
5
\frac{2 \pi}{5}
5
2
π
, find the corresponding angle in Quadrant
2
2
2
.
\newline
Answer:
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Solve for the exact value of
x
x
x
.
\newline
2
ln
(
2
x
−
2
)
+
12
=
8
2 \ln (2 x-2)+12=8
2
ln
(
2
x
−
2
)
+
12
=
8
\newline
Answer:
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\newline
The angle
θ
1
\theta_{1}
θ
1
is located in Quadrant
I
\mathrm{I}
I
, and
sin
(
θ
1
)
=
17
20
\sin \left(\theta_{1}\right)=\frac{17}{20}
sin
(
θ
1
)
=
20
17
.
\newline
What is the value of
cos
(
θ
1
)
\cos \left(\theta_{1}\right)
cos
(
θ
1
)
?
\newline
Express your answer exactly.
\newline
cos
(
θ
1
)
=
\cos \left(\theta_{1}\right)=
cos
(
θ
1
)
=
Get tutor help
If
cos
(
θ
)
=
8
17
\cos(\theta) = \frac{8}{17}
cos
(
θ
)
=
17
8
and
0
∘
<
θ
<
9
0
∘
0^\circ < \theta < 90^\circ
0
∘
<
θ
<
9
0
∘
, what is
sec
(
θ
)
\sec(\theta)
sec
(
θ
)
?
\newline
Write your answer in simplified, rationalized form.
\newline
sec
(
θ
)
=
\sec(\theta) =
sec
(
θ
)
=
______
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Given
sec
A
=
61
6
\sec A = \frac{\sqrt{61}}{6}
sec
A
=
6
61
and that angle
A
A
A
is in Quadrant I, find the exact value of
csc
A
\csc A
csc
A
in simplest radical form using a rational denominator.
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E
=
[
0
3
5
5
5
2
]
\mathrm{E}=\left[\begin{array}{lll}0 & 3 & 5 \\ 5 & 5 & 2\end{array}\right]
E
=
[
0
5
3
5
5
2
]
and
D
=
[
3
4
3
−
2
4
−
2
]
\mathrm{D}=\left[\begin{array}{rr}3 & 4 \\ 3 & -2 \\ 4 & -2\end{array}\right]
D
=
⎣
⎡
3
3
4
4
−
2
−
2
⎦
⎤
\newline
Let
H
=
E
D
\mathrm{H}=\mathrm{ED}
H
=
ED
. Find
H
\mathrm{H}
H
.
\newline
H
=
\mathbf{H}=
H
=
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Find the midpoint
m
m
m
of
z
1
=
(
7
+
8
i
)
z_{1}=(7+8 i)
z
1
=
(
7
+
8
i
)
and
z
2
=
(
8
−
7
i
)
z_{2}=(8-7 i)
z
2
=
(
8
−
7
i
)
.
\newline
Express your answer in rectangular form.
\newline
m
=
□
m=\square
m
=
□
Get tutor help
Express
z
1
=
6
+
2
3
i
z_{1}=6+2 \sqrt{3} i
z
1
=
6
+
2
3
i
in polar form.
\newline
Express your answer in exact terms, using degrees, where your angle is between
0
∘
0^{\circ}
0
∘
and
36
0
∘
360^{\circ}
36
0
∘
, inclusive.
\newline
z
1
=
z_{1}=
z
1
=
Get tutor help
Express
z
1
=
−
8
3
+
8
i
z_{1}=-8 \sqrt{3}+8 i
z
1
=
−
8
3
+
8
i
in polar form.
\newline
Express your answer in exact terms, using degrees, where your angle is between
0
∘
0^{\circ}
0
∘
and
36
0
∘
360^{\circ}
36
0
∘
, inclusive.
\newline
z
1
=
z_{1}=
z
1
=
Get tutor help
z
=
5
−
3
i
z=5-3 i
z
=
5
−
3
i
\newline
Find the angle
θ
\theta
θ
(in degrees) that
z
z
z
makes in the complex plane.
\newline
Round your answer, if necessary, to the nearest tenth. Express
θ
\theta
θ
between
−
18
0
∘
-180^{\circ}
−
18
0
∘
and
18
0
∘
180^{\circ}
18
0
∘
.
\newline
θ
=
□
∘
\theta=\square^{\circ}
θ
=
□
∘
Get tutor help
z
=
1
−
2
i
z=1-2 i
z
=
1
−
2
i
\newline
Find the angle
θ
\theta
θ
(in degrees) that
z
z
z
makes in the complex plane.
\newline
Round your answer, if necessary, to the nearest tenth. Express
θ
\theta
θ
between
−
18
0
∘
-180^{\circ}
−
18
0
∘
and
18
0
∘
180^{\circ}
18
0
∘
.
\newline
θ
=
□
∘
\theta=\square^{\circ}
θ
=
□
∘
Get tutor help
z
=
1
+
4
i
z=1+4 i
z
=
1
+
4
i
\newline
Find the angle
θ
\theta
θ
(in degrees) that
z
z
z
makes in the complex plane.
\newline
Round your answer, if necessary, to the nearest tenth. Express
θ
\theta
θ
between
−
18
0
∘
-180^{\circ}
−
18
0
∘
and
18
0
∘
180^{\circ}
18
0
∘
.
\newline
θ
=
□
∘
\theta=\square^{\circ}
θ
=
□
∘
Get tutor help
z
=
3
−
4
i
z=3-4 i
z
=
3
−
4
i
\newline
Find the angle
θ
\theta
θ
(in radians) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express
θ
\theta
θ
between
−
π
-\pi
−
π
and
π
\pi
π
.
\newline
θ
=
\theta=
θ
=
Get tutor help
z
=
−
6
−
5
i
z=-6-5 i
z
=
−
6
−
5
i
\newline
Find the angle
θ
\theta
θ
(in degrees) that
z
z
z
makes in the complex plane.
\newline
Round your answer, if necessary, to the nearest tenth. Express
θ
\theta
θ
between
−
18
0
∘
-180^{\circ}
−
18
0
∘
and
18
0
∘
180^{\circ}
18
0
∘
.
\newline
θ
=
□
∘
\theta=\square^{\circ}
θ
=
□
∘
Get tutor help
z
=
8
−
9
i
z=8-9 i
z
=
8
−
9
i
\newline
Find the angle
θ
\theta
θ
(in radians) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express
θ
\theta
θ
between
−
π
-\pi
−
π
and
π
\pi
π
.
\newline
θ
=
\theta=
θ
=
Get tutor help
z
=
4
+
3
i
z=4+3 i
z
=
4
+
3
i
\newline
Find the angle
θ
\theta
θ
(in radians) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express
θ
\theta
θ
between
−
π
-\pi
−
π
and
π
\pi
π
.
\newline
θ
=
\theta=
θ
=
Get tutor help
z
=
−
5
+
3
i
z=-5+3 i
z
=
−
5
+
3
i
\newline
Find the angle
θ
\theta
θ
(in degrees) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest tenth. Express
θ
\theta
θ
between
−
18
0
∘
-180^{\circ}
−
18
0
∘
and
18
0
∘
180^{\circ}
18
0
∘
.
\newline
θ
=
□
∘
\theta=\square^{\circ}
θ
=
□
∘
Get tutor help
z
=
−
7
−
6
i
z=-7-6 i
z
=
−
7
−
6
i
\newline
Find the angle
θ
\theta
θ
(in degrees) that
z
z
z
makes in the complex plane.
\newline
Round your answer, if necessary, to the nearest tenth. Express
θ
\theta
θ
between
−
18
0
∘
-180^{\circ}
−
18
0
∘
and
18
0
∘
180^{\circ}
18
0
∘
.
\newline
θ
=
□
∘
\theta=\square^{\circ}
θ
=
□
∘
Get tutor help
z
=
−
3
−
8
i
z=-3-8 i
z
=
−
3
−
8
i
\newline
Find the angle
θ
\theta
θ
(in degrees) that
z
z
z
makes in the complex plane.
\newline
Round your answer, if necessary, to the nearest tenth. Express
θ
\theta
θ
between
−
18
0
∘
-180^{\circ}
−
18
0
∘
and
18
0
∘
180^{\circ}
18
0
∘
.
\newline
θ
=
□
∘
\theta=\square^{\circ}
θ
=
□
∘
Get tutor help
z
=
−
8
+
5
i
z=-8+5 i
z
=
−
8
+
5
i
\newline
Find the angle
θ
\theta
θ
(in degrees) that
z
z
z
makes in the complex plane.
\newline
Round your answer, if necessary, to the nearest tenth. Express
θ
\theta
θ
between
−
18
0
∘
-180^{\circ}
−
18
0
∘
and
18
0
∘
180^{\circ}
18
0
∘
.
\newline
θ
=
□
∘
\theta=\square^{\circ}
θ
=
□
∘
Get tutor help
z
=
7
+
3
i
z=7+3 i
z
=
7
+
3
i
\newline
Find the angle
θ
\theta
θ
(in degrees) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest tenth. Express
θ
\theta
θ
between
−
18
0
∘
-180^{\circ}
−
18
0
∘
and
18
0
∘
180^{\circ}
18
0
∘
.
\newline
θ
=
□
∘
\theta=\square^{\circ}
θ
=
□
∘
Get tutor help
z
=
6
+
5
i
z=6+5 i
z
=
6
+
5
i
\newline
Find the angle
θ
\theta
θ
(in radians) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express
θ
\theta
θ
between
−
π
-\pi
−
π
and
π
\pi
π
.
\newline
θ
=
\theta=
θ
=
Get tutor help
z
=
5
+
7
i
z=5+7 i
z
=
5
+
7
i
\newline
Find the angle
θ
\theta
θ
(in radians) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express
θ
\theta
θ
between
−
π
-\pi
−
π
and
π
\pi
π
.
\newline
θ
=
\theta=
θ
=
Get tutor help
z
=
−
8
+
3
i
z=-8+3 i
z
=
−
8
+
3
i
\newline
Find the angle
θ
\theta
θ
(in radians) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express
θ
\theta
θ
between
−
π
-\pi
−
π
and
π
\pi
π
.
\newline
θ
=
\theta=
θ
=
Get tutor help
z
=
−
2
+
8
i
z=-2+8 i
z
=
−
2
+
8
i
\newline
Find the angle
θ
\theta
θ
(in radians) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express
θ
\theta
θ
between
−
π
-\pi
−
π
and
π
\pi
π
.
\newline
θ
=
\theta=
θ
=
Get tutor help
z
=
4
−
2
i
z=4-2 i
z
=
4
−
2
i
\newline
Find the angle
θ
\theta
θ
(in degrees) that
z
z
z
makes in the complex plane.
\newline
Round your answer, if necessary, to the nearest tenth. Express
θ
\theta
θ
between
−
18
0
∘
-180^{\circ}
−
18
0
∘
and
18
0
∘
180^{\circ}
18
0
∘
.
\newline
θ
=
□
∘
\theta=\square^{\circ}
θ
=
□
∘
Get tutor help
A complex number
z
1
z_{1}
z
1
has a magnitude
∣
z
1
∣
=
4
\left|z_{1}\right|=4
∣
z
1
∣
=
4
and an angle
θ
1
=
33
0
∘
\theta_{1}=330^{\circ}
θ
1
=
33
0
∘
.
\newline
Express
z
1
z_{1}
z
1
in rectangular form, as
z
1
=
a
+
b
i
z_{1}=a+b i
z
1
=
a
+
bi
.
\newline
Express
a
+
b
i
a+b i
a
+
bi
in exact terms.
\newline
z
1
=
□
z_{1}=\square
z
1
=
□
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z
=
−
3
−
6
i
z=-3-6 i
z
=
−
3
−
6
i
\newline
Find the angle
θ
\theta
θ
(in radians) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express
θ
\theta
θ
between
−
π
-\pi
−
π
and
π
\pi
π
.
\newline
θ
=
\theta=
θ
=
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z
=
−
3
+
5
i
z=-3+5 i
z
=
−
3
+
5
i
\newline
Find the angle
θ
\theta
θ
(in radians) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express
θ
\theta
θ
between
−
π
-\pi
−
π
and
π
\pi
π
.
\newline
θ
=
\theta=
θ
=
Get tutor help
z
=
−
2
−
5
i
z=-2-5 i
z
=
−
2
−
5
i
\newline
Find the angle
θ
\theta
θ
(in radians) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express
θ
\theta
θ
between
−
π
-\pi
−
π
and
π
\pi
π
.
\newline
θ
=
\theta=
θ
=
Get tutor help
The following are all angle measures (in degrees, rounded to the nearest tenth) whose sine is
0
0
0
.
36
36
36
.
\newline
Which is the principal value of
arcsin
(
0.36
)
\arcsin (0.36)
arcsin
(
0.36
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
21.
1
∘
21.1^{\circ}
21.
1
∘
\newline
(B)
381.
1
∘
381.1^{\circ}
381.
1
∘
\newline
(C)
741.
1
∘
741.1^{\circ}
741.
1
∘
\newline
(D)
1101.
1
∘
1101.1^{\circ}
1101.
1
∘
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The following are all angle measures (in degrees, rounded to the nearest tenth) whose sine is
−
0
-0
−
0
.
71
71
71
.
\newline
Which is the principal value of
arcsin
(
−
0.71
)
\arcsin (-0.71)
arcsin
(
−
0.71
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
405.
2
∘
-405.2^{\circ}
−
405.
2
∘
\newline
(B)
−
45.
2
∘
-45.2^{\circ}
−
45.
2
∘
\newline
(C)
314.
8
∘
314.8^{\circ}
314.
8
∘
\newline
(D)
674.
8
∘
674.8^{\circ}
674.
8
∘
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The following formula gives the volume
V
V
V
of a pyramid, where
A
A
A
is the area of the base and
h
h
h
is the height:
\newline
V
=
1
3
A
h
V=\frac{1}{3} A h
V
=
3
1
A
h
\newline
Rearrange the formula to highlight the base area.
\newline
A
=
A=
A
=
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The following formula gives an object's kinetic energy
K
K
K
, where
m
m
m
is the object's mass and
v
v
v
is the object's velocity.
\newline
K
=
1
2
m
v
2
K=\frac{1}{2} m v^{2}
K
=
2
1
m
v
2
\newline
Rearrange the formula to highlight mass.
\newline
m
=
m=
m
=
Get tutor help
A
=
π
r
2
θ
360
A=\frac{\pi r^{2} \theta}{360}
A
=
360
π
r
2
θ
\newline
The given equation can be used to find the area,
A
A
A
, of a sector of a circle of radius,
r
r
r
, where
θ
\theta
θ
is the sector's central angle in degrees. Which of the following correctly shows the circle sector's radius in terms of the area of the sector and the central angle?
\newline
Choose
1
1
1
answer:
\newline
(A)
r
=
360
A
π
θ
r=\sqrt{\frac{360 A}{\pi \theta}}
r
=
π
θ
360
A
\newline
(B)
r
=
360
A
π
r
θ
r=\frac{360 A}{\pi r \theta}
r
=
π
r
θ
360
A
\newline
(C)
r
=
π
θ
360
A
r=\sqrt{\frac{\pi \theta}{360 A}}
r
=
360
A
π
θ
\newline
(D)
r
=
π
θ
360
A
r
r=\frac{\pi \theta}{360 A r}
r
=
360
A
r
π
θ
Get tutor help
V
=
π
r
2
h
V=\pi r^{2} h
V
=
π
r
2
h
\newline
The equation gives the Volume
V
V
V
of a right cylinder with radius
r
r
r
and height
h
h
h
. Which of the following equations correctly gives the radius of the cylinder in terms of the cylinder's volume and height?
\newline
Choose
1
1
1
answer:
\newline
(A)
r
=
V
h
π
r=\frac{\sqrt{V h}}{\pi}
r
=
π
Vh
\newline
(B)
r
=
V
h
π
r=\sqrt{\frac{V h}{\pi}}
r
=
π
Vh
\newline
(C)
r
=
V
π
h
r=\frac{\sqrt{V}}{\pi h}
r
=
πh
V
\newline
(D)
r
=
V
π
h
r=\sqrt{\frac{V}{\pi h}}
r
=
πh
V
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