OK

[Lizzie]

understanding this problem will help me understand the next like 3 problems on this page, so any help would be appreciated.

The volume of a cylinder is V=Pi r²h, (where Pi=3.14159). If dr/dt=-4 and dh/dt=10, then when r=1 and h=1, the volume of the cylinder is:

ok, so how do I do about doing this? It's probably quite simple, but you all know me. Takes a bit of pointing to get me going. I'm thinking that I am supposed to get the equation for dh/dt and the equation for dh/dt and figure our what's going on by plugging in r=1 and h=1??

The question asks if the volume is increasing, decreasing, remaining constant, or not enough information to tell... Is the proper amount of information given??

 

[pka]

Think of each as function of t: \(\displaystyle V(t) = \pi r^2 (t)h(t)\).
Also realize that \(\displaystyle r^2 (t)h(t)\) is a product and must be differentiated as such.

Now you give it a try.

 

[Lizzie]

Where the heck does t come in anyway? lol, is t supposed to be V? Argh! but then t can't be V because V(t) doesn't make any sense, so is t just a number?

 

[pka]

They are functions of TIME!
You are given \(\displaystyle \frac{{dr}}{{dt}} = - 4\), where did that t come from! AND \(\displaystyle \frac{{dh}}{{dt}} = 10\).

Find \(\displaystyle \frac{{dV}}{{dt}}\). Then use the given with r=1 and h=1.

 

[Daniel_Feldman]

Nope...pka merely expresser r, v, and h as functions of t. Remember how, in algebra, you had functions in the form f(x) (and still do have them in Calculus!)? You weren't using two variables there, were you? It was merely a way of expressing the function.

First of all, the first part of the problem involves no calculus at all. It merely requires substitution.

V=Pi r^2h

All it asks for is the volume(V) when r=1 and h=1.

V=pi(1)^2(1)=pi



The second part is where it becomes important to understand the concept of related rates. With these sorts of problems, it's important to know two things:

a. What the problem is asking.
b. What you are differentiating with respect to.


"The question asks if the volume is increasing, decreasing, remaining constant, or not enough information to tell."

Ok. So in other words, it's asking if the change in volume is positive, negative, 0, or whether we're not given enough info to tell.

Let's note that we are looking for the change in volume as time changes; That is, we are differentiating with respect to time, so we must find dV/dt.

V=pi r^2h
dV/dt=pi(r^2 dh/dt +2rh dr/dt)

See how I did that? It's the product rule!

Next, we simply substitute.

dV/dt=pi(r^2 dh/dt +2rh dr/dt)

=pi((1)^2(10)+2(1)(1)(-4))
=pi(10-8)
=2pi

2pi is greater than 0, so the volume is increasing.

Any questions?

 

[Lizzie]

wow, that was extremely helpful! Thank you very much!.