Step-by-step explanation:Let us set up the following variables:
enter image source here
Radius of sphere at time t
(cm)
A
Surface area of sphere at time t
(cm
2
)
V
Volume of sphere at time t
(cm
3
)
t
time
(sec)
Our aim is to find
d
A
d
t
when
V
=
9
Ο
2
and
d
V
d
t
=
Ο
3
.
The standard formula for Area & Volume of a sphere are:
V
=
4
3
Ο
r
3
...
.
[
1
]
A
=
4
Ο
r
2
...
.
[
2
]
When
V
=
9
Ο
2
β
4
3
Ο
r
3
=
9
Ο
2
β΄
r
3
=
9
2
β
3
4
β΄
r
=
3
2
Differentiating [1] and [2] wrt
r
we get;
d
V
d
r
=
4
Ο
r
2
and
d
A
d
r
=
8
Ο
r
And from the chain rule we get:
d
A
d
t
=
d
A
d
r
β
d
r
d
V
β
d
V
d
t
=
8
Ο
r
β
1
4
Ο
r
2
β
d
V
d
t
=
2
r
β
d
V
d
t
So when
V
=
9
Ο
2
,
d
V
d
t
=
Ο
3
and
r
=
3
2
, then:
d
A
d
t
=
2
3
2
β
Ο
3
=
4
Ο
9
Answer link
salamat
Feb 20, 2017
4
9
Ο
cm^2/s
Explanation:
V
=
4
3
Ο
β
r
3
, where V =volume of sphere and r =radius.
given that,
d
V
d
t
=
Ο
3
and
V
=
9
2
Ο
4
3
Ο
β
r
3
=
9
2
Ο
r
3
=
9
2
Ο
β
3
4
Ο
r
3
=
27
8
,
r
=
3
2
V
=
4
3
Ο
β
r
3
,
then
d
V
d
r
=
4
Ο
β
r
2
when
r
=
3
2
,
d
V
d
r
=
4
Ο
β
(
3
2
)
2
=
9
Ο
d
r
d
t
=
d
V
d
t
/
(
d
V
d
r
)
=
Ο
3
9
Ο
=
1
27
Area pf sphere,
A
=
4
Ο
r
2
d
A
d
r
=
8
Ο
r
when
r
=
3
2
,
d
A
d
r
=
8
Ο
r
=
8
Ο
(
3
2
)
=
12
Ο
therefore,
d
A
d
t
=
d
A
d
r
β
d
r
d
t
=
12
Ο
β
1
27
=
4
9
Ο
(
c
m
)
2
s
Answer link
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Let us set up the following variables:
enter image source here
β§
βͺ
βͺ
βͺ
βͺ
βͺ
β¨
βͺ
βͺ
βͺ
βͺ
βͺ
β©
r
Radius of sphere at time t
(cm)
A
Surface area of sphere at time t
(cm
2
)
V
Volume of sphere at time t
(cm
3
)
t
time
(sec)
Our aim is to find
d
A
d
t
when
V
=
9
Ο
2
and
d
V
d
t
=
Ο
3
.
The standard formula for Area & Volume of a sphere are:
V
=
4
3
Ο
r
3
...
.
[
1
]
A
=
4
Ο
r
2
...
.
[
2
]
When
V
=
9
Ο
2
β
4
3
Ο
r
3
=
9
Ο
2
β΄
r
3
=
9
2
β
3
4
β΄
r
=
3
2
Differentiating [1] and [2] wrt
r
we get;
d
V
d
r
=
4
Ο
r
2
and
d
A
d
r
=
8
Ο
r
And from the chain rule we get:
d
A
d
t
=
d
A
d
r
β
d
r
d
V
β
d
V
d
t
=
8
Ο
r
β
1
4
Ο
r
2
β
d
V
d
t
=
2
r
β
d
V
d
t
So when
V
=
9
Ο
2
,
d
V
d
t
=
Ο
3
and
r
=
3
2
, then:
d
A
d
t
=
2
3
2
β
Ο
3
=
4
Ο
9
Answer link
salamat
Feb 20, 2017
4
9
Ο
cm^2/s
Explanation:
V
=
4
3
Ο
β
r
3
, where V =volume of sphere and r =radius.
given that,
d
V
d
t
=
Ο
3
and
V
=
9
2
Ο
4
3
Ο
β
r
3
=
9
2
Ο
r
3
=
9
2
Ο
β
3
4
Ο
r
3
=
27
8
,
r
=
3
2
V
=
4
3
Ο
β
r
3
,
then
d
V
d
r
=
4
Ο
β
r
2
when
r
=
3
2
,
d
V
d
r
=
4
Ο
β
(
3
2
)
2
=
9
Ο
d
r
d
t
=
d
V
d
t
/
(
d
V
d
r
)
=
Ο
3
9
Ο
=
1
27
Area pf sphere,
A
=
4
Ο
r
2
d
A
d
r
=
8
Ο
r
when
r
=
3
2
,
d
A
d
r
=
8
Ο
r
=
8
Ο
(
3
2
)
=
12
Ο
therefore,
d
A
d
t
=
d
A
d
r
β
d
r
d
t
=
12
Ο
β
1
27
=
4
9
Ο
(
c
m
)
2
s
Answer link
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How do you determine the rate of change of a function?
What are partial derivatives?
How do you find a function f(x), which, when multiplied by its derivative, gives you
x
3
, and...
How do you graph the derivative of a function when you are given the graph of the function?
What is the purpose of a derivative?
How do you solve the AP Calculus 2013 Free Response question...
A factory produces bicycles at a rate of 80+0.5t^2-0.7t bicycles per week (t in weeks). How...
The cost function for a product is C(x)=0.8x^2 +120x+110. How to find average cost over [0,600] ?
A ladder 10ft long rests against a vertical wall. If the bottom of the ladder slides away from...
See all questions in Rate of Change of a Function
Impact of this question
11315 views around the world
You can reuse this answer
Creative Commons License
