Rate of change of area in sphereical baloon, .

[christian0710]

Homework Statement

I have a homework assignment that I find challengingA spherical balloon is being inflated at a rate of 10 cu.in/sec (i assume it's cubic inches per second)

Find the rate of change of area when the balloon has radius=6

Homework Equations

So far I know that

V=(4/3)*pi*r^3
A=4*pi*r^2

dV/dt = 10 in^3/secSo the question is, how do I find dA/dt when radius=6 and the volume increases at a rate of 10 in^3/sec ?

I'm realyl stuck here. Can someone help?

The Attempt at a Solution

I've just been wild guessing here. I know that dV/dt= 4*pi*r*dr/dt and that dV/dt=10, but if i substitute it into
10= 4*pi*r*dr/dt and isolate dr/dt, I don't really get anywhere.

The answer is supposed to be 10/3 in^2 sec.

 

[Dr. Courtney]

Apply the chain rule more carefully. You have some errors.

I find it helpful to slow down and take more careful (smaller steps):

dV/dt = dV/dr * dr/dt

dV/dr = stuff
...

 

[christian0710]

So dV/dr = 4*pi*r^2
dr/dt = How do i find this?

 

[Dr. Courtney]

So dV/dr = 4*pi*r^2
dr/dt = How do i find this?

It's the only unknown left in the dV/dr equation.

Solve for it as a number, then plug that number into your equation for dA/dt = dA/dr * dr/dt.

 

[RUber]

You are given a rate of inflation, this is dV/dt. First you say it is 6 but in your work you refer to 10. Please be more clear.
You have the stuff you need already.
dV/dt = dV/dr * dr/dt
dA/dt = dA/dr * dr/dt

You can simply use the given information to find dr/dt, then plug that into your equation for change in area.

[edit] I was too late.

 

[christian0710]

Thank you so much, it works!

just a last question: So how do you just know that you must apply the 2 chain rules beneath?

dV/dt = dV/dr * dr/dt
dA/dt = dA/dr * dr/dt.

Is it beacuse you know that the change in V and A depends on time and on radius r?

 

[RUber]

It is because the change in r depends on t.
If G(a) = G(a(t)) then dG/dt = dG/da*da/dt. That is what the chain rule says.

 

[Dr. Courtney]

I knew what to do from years of teaching Calculus and for the reasons RUber described.

Students may not always see what to do, so I encourage them:

When you do not know what to do, take some steps you do know how to execute properly.

Then reassess and see if you are any closer to finding the variable you are trying to solve for.

 

[christian0710]

Thank you very much! I really appreciate the help !