The volume of a sphere is increasing at a constant rate of 2952 cubic centimeters per second. At the instant when the radius of the sphere is 1010 centimeters, what is the rate of change of the radius? The volume of a sphere can be found with the equation V=\frac{4}{3}\pi r^3.V= 3
4

πr
3
. Round your answer to three decimal places.

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

π

2

and

d

V

d

t

=

π

3

.

The standard formula for Area & Volume of a sphere are:

V

=

4

3

π

r

3

...

.

[

1

]

A

=

4

π

r

2

...

.

[

2

]

When

V

=

9

π

2

⇒

4

3

π

r

3

=

9

π

2

∴

r

3

=

9

2

⋅

3

4

∴

r

=

3

2

Differentiating [1] and [2] wrt

r

we get;

d

V

d

r

=

4

π

r

2

and

d

A

d

r

=

8

π

r

And from the chain rule we get:

d

A

d

t

=

d

A

d

r

⋅

d

r

d

V

⋅

d

V

d

t

=

8

π

r

⋅

1

4

π

r

2

⋅

d

V

d

t

=

2

r

⋅

d

V

d

t

So when

V

=

9

π

2

,

d

V

d

t

=

π

3

and

r

=

3

2

, then:

d

A

d

t

=

2

3

2

⋅

π

3

=

4

π

9

Answer link

salamat

Feb 20, 2017

4

9

π

cm^2/s

Explanation:

V

=

4

3

π

⋅

r

3

, where V =volume of sphere and r =radius.

given that,

d

V

d

t

=

π

3

and

V

=

9

2

π

4

3

π

⋅

r

3

=

9

2

π

r

3

=

9

2

π

⋅

3

4

π

r

3

=

27

8

,

r

=

3

2

V

=

4

3

π

⋅

r

3

,

then

d

V

d

r

=

4

π

⋅

r

2

when

r

=

3

2

,

d

V

d

r

=

4

π

⋅

(

3

2

)

2

=

9

π

d

r

d

t

=

d

V

d

t

/

(

d

V

d

r

)

=

π

3

9

π

=

1

27

Area pf sphere,

A

=

4

π

r

2

d

A

d

r

=

8

π

r

when

r

=

3

2

,

d

A

d

r

=

8

π

r

=

8

π

(

3

2

)

=

12

π

therefore,

d

A

d

t

=

d

A

d

r

⋅

d

r

d

t

=

12

π

⋅

1

27

=

4

9

π

(

c

m

)

2

s

Answer link

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Let us set up the following variables:

enter image source here

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩

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

π

2

and

d

V

d

t

=

π

3

.

The standard formula for Area & Volume of a sphere are:

V

=

4

3

π

r

3

...

.

[

1

]

A

=

4

π

r

2

...

.

[

2

]

When

V

=

9

π

2

⇒

4

3

π

r

3

=

9

π

2

∴

r

3

=

9

2

⋅

3

4

∴

r

=

3

2

Differentiating [1] and [2] wrt

r

we get;

d

V

d

r

=

4

π

r

2

and

d

A

d

r

=

8

π

r

And from the chain rule we get:

d

A

d

t

=

d

A

d

r

⋅

d

r

d

V

⋅

d

V

d

t

=

8

π

r

⋅

1

4

π

r

2

⋅

d

V

d

t

=

2

r

⋅

d

V

d

t

So when

V

=

9

π

2

,

d

V

d

t

=

π

3

and

r

=

3

2

, then:

d

A

d

t

=

2

3

2

⋅

π

3

=

4

π

9

Answer link

salamat

Feb 20, 2017

4

9

π

cm^2/s

Explanation:

V

=

4

3

π

⋅

r

3

, where V =volume of sphere and r =radius.

given that,

d

V

d

t

=

π

3

and

V

=

9

2

π

4

3

π

⋅

r

3

=

9

2

π

r

3

=

9

2

π

⋅

3

4

π

r

3

=

27

8

,

r

=

3

2

V

=

4

3

π

⋅

r

3

,

then

d

V

d

r

=

4

π

⋅

r

2

when

r

=

3

2

,

d

V

d

r

=

4

π

⋅

(

3

2

)

2

=

9

π

d

r

d

t

=

d

V

d

t

/

(

d

V

d

r

)

=

π

3

9

π

=

1

27

Area pf sphere,

A

=

4

π

r

2

d

A

d

r

=

8

π

r

when

r

=

3

2

,

d

A

d

r

=

8

π

r

=

8

π

(

3

2

)

=

12

π

therefore,

d

A

d

t

=

d

A

d

r

⋅

d

r

d

t

=

12

π

⋅

1

27

=

4

9

π

(

c

m

)

2

s

Answer link

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What is Rate of Change of a Function?

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

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11315 views around the world

You can reuse this answer

Creative Commons License

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step-by-step explanation: