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Let’s check out your problem:
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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Math Problems
Algebra 2
Write and solve direct variation equations
Full solution
Q.
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.
Identify Relationship:
Identify the relationship and differentiate implicitly.
\newline
Given
y
=
π
x
2
y = \pi x^2
y
=
π
x
2
, differentiate both sides with respect to
t
t
t
.
\newline
d
y
d
t
=
2
π
x
(
d
x
d
t
)
\frac{dy}{dt} = 2\pi x\left(\frac{dx}{dt}\right)
d
t
d
y
=
2
π
x
(
d
t
d
x
)
Differentiate Implicitly:
Substitute the values of
d
x
d
t
\frac{dx}{dt}
d
t
d
x
and
x
x
x
.
\newline
d
x
d
t
=
−
1
8
\frac{dx}{dt} = -\frac{1}{8}
d
t
d
x
=
−
8
1
and
x
=
16
x = 16
x
=
16
.
\newline
d
y
d
t
=
2
π
(
16
)
(
−
1
8
)
\frac{dy}{dt} = 2\pi(16)(-\frac{1}{8})
d
t
d
y
=
2
π
(
16
)
(
−
8
1
)
Substitute Values:
Simplify the expression to find
d
y
d
t
\frac{dy}{dt}
d
t
d
y
.
d
y
d
t
=
2
π
(
16
)
(
−
1
8
)
=
−
4
π
\frac{dy}{dt} = 2\pi(16)(-\frac{1}{8}) = -4\pi
d
t
d
y
=
2
π
(
16
)
(
−
8
1
)
=
−
4
π
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\newline
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\newline
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\newline
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\newline
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\[[C]a = \frac{k}{bcd}\]
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\newline
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(
A
)
(
B
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(
B
)
(
A
)
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A
A
A
and
B
B
B
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\newline
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\newline
Write your answer as an equation with
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\newline
Write your answer as an equation with
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first, followed by an equals sign.
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