# Find the derivative of a balloon's circumference

A balloon's volume is increasing at a rate of dV/dt. Express the rate of change of the circumference with respect to time (dc/dt) in terms of the volume and radius.

## Homework Equations

Vsphere = (4/3)(π)(r^3)
C = (2)(π)(r)

## The Attempt at a Solution

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My strategy was to come up with two separate equations of dr/dt and substitute leading to an equation relating dv/dt and dc/dt

dV/dt = (4)(π)(r^2) (dr/dt)
dc/dt = (2)(π)(dr/dt)

(dc/dt) (1/2π) = dr/dt
Substitute...

(4)(π)(r^2) (dc/dt) (1/2π) = dv/dt

Canceling like terms

(2r^2)(dc/dt) = dv/dt

(1/2r^2) (dv/dt) = (dc/dt)

Now I was curious and wanted to graph the function just for fun. I know dv/dt is some number so I assumed it is 3. This yields
3/2r^2 = dc/dt
If I were to graph this function on a graphing utility would I be getting the circumference on the y axis and time on the x? I ask this because the function is decreasing but the circumference of the balloon is increasing. Please help.

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kuruman
Homework Helper
Gold Member
Where did you get this idea? dC/dt is positive which means that C increases as time increases. The r2 in the denominator means that C is increasing at a decreasing rate as the balloon grows larger, not that C is decreasing.
If you were to graph this, you need to integrate first to find C as a function of time. You would plot C on the vertical axis and t on the horizontal axis.

Last edited:
Clear and concise answer much thanks. But I'm currently in a pre-calculus class and we have not gone over integration yet. If it is not too much trouble can you please show how to do it to obtain circumference as a function of time? Thanks once again

kuruman
Homework Helper
Gold Member
Clear and concise answer much thanks. But I'm currently in a pre-calculus class and we have not gone over integration yet. If it is not too much trouble can you please show how to do it to obtain circumference as a function of time? Thanks once again
Is the statement of the problem exactly as was given to you? If so, then you cannot figure out what C as a function of time looks like because you need to know what dV/dt is, i.e. how the volume is increasing. You assumed as an example that it is 3, but is it really? Furthermore, is dV/dt constant or does itself depend on time? So if dV/dt is not given, you cannot do anything more than you have already done.

Mark44
Mentor
But I'm currently in a pre-calculus class and we have not gone over integration yet.
Thread moved. Problems involving derivatives are definitely in the bailiwick of calculus, regardless of whether the class you are taking is called "precalculus."
Please post problems involving derivatives and related rates in the Calculus & Beyond Homework section - thanks...