Struggling with a rate of change problem.

A spherical snowball is melting in such a way that its diameter is decreasing at rate of 3 cm/min. At what rate is the volume of the snowball decreasing when the diameter is 11cm?

When the diameter is 11cm, the volume of the snowball is decreasing at a rate of _______ .

I know that the volume of a sphere is . So I use implicit differentiation and take the derivative of both sides and use the chain rule because I know is a function of , and I believe this rate is given as -3, so .

Since the radius of a circle is one half the diameter... , however, this answer is incorrect.

I was given a hint that I have to write an equation linking the changing volume V(t) to the changing diameter d(t), where

I don't understand how to figure this out. Also, I don't have the answer so please don't give it to me. Thanks in advance.

Re: Struggling with a rate of change problem.

Hi Khan! Well I'm not really certain of this solution but give it a try(Wink):

1)State the given:

, , so we attempt to find

2)Derive a formula for Volume as a function of diameter:

Volume of a sphere is given as , but since , then

Then we plug r back in to get:

So that by simplifying, we get:

3)Take the derivative of the equation with respect to time:

We get:

Take it from here dude. Remember that you only have to plug in all the values that is on part 1 of this solution. And since it is a negative rate of change :D

Re: Struggling with a rate of change problem.

--- minut errror gimme a min, nvm the above poster has right answer. cheers

Re: Struggling with a rate of change problem.

Wow did I hehe! Thanks man that really boosted me up on doing this integrals early in the morning:D! Anyways dude, I saw your solution, and for some reason you used partial derivatives. Damn, if that worked I would love you to educated me more about it. Cheers!

Re: Struggling with a rate of change problem.

It did work (As for the partial derivative i dont know how to use latex for normal derivative so i erronously use the partial symbol, its a very bad habit i know!), but i didnt read his question correctly so i made it much more complicated than it had to be, so instead of fixing it, i saw your very correct solution and said, bah humbug.

Re: Struggling with a rate of change problem.

Aww man, what a bummer. I thought someone was finally able to solve it using partial derivatives:D. Either way, nice try big man!

Re: Struggling with a rate of change problem.

I'm not sure where you would use partial derivatives in this case as Volume is a function of one variable D (which itself is a function of variable t) ultimately it is still a function of one variable whether it is r or d or t. If he had said the diamater varied according to time and some other variable, then we could possible use partial derivatives.

Re: Struggling with a rate of change problem.

Exactly my point! So if you are able to solve it using partial derivatives then you are my new hero! :D

Re: Struggling with a rate of change problem.

But when there is one variable the partial derivate become the normal derivative no? So i can solve it using partial derivatives!.

Re: Struggling with a rate of change problem.

Thank you for your help, but I don't understand why plugging (which is what the radius) into the equation does not work.

The equations ,and look equivalent to me in that they both should solve the problem. In other words, how is the equation different from the equation before and after they are differentiated. Thanks.

Re: Struggling with a rate of change problem.

Quote:

Originally Posted by

**KhanDisciple** Thank you for your help, but I don't understand why plugging

(which is what the radius) into the equation

does not work.

The equations

,and

look equivalent to me in that they both should solve the problem. In other words, how is the equation

different from the equation

before and after they are differentiated. Thanks.

you are correct also. They are indeed equivalent. and doing it your way and plugging it for r and for will also give you the rate at which volume is changing with time, when diameter is 11.

Re: Struggling with a rate of change problem.

Thank you for your clarification guys, I really appreciate it.