# Thread: Calc IV: Functions of Several Variables (Chain Rule)

1. ## Calc IV: Functions of Several Variables (Chain Rule)

Hello,
I am having a hard time solving this problem. I have no clue. None whatsover. Thanks in advance for your help.

Volume & Surface Area
The radius of a right circular cone is increasing at a rate of 6 inches per minute, and the height is decreasing at a rate of 4 inches per minute. What are the rates of change of the volume and surface area when radius is 12 inches and height is 36 inches?

2. Hello,

$V=(\pi r^2)\frac h3=\frac{\pi}{3} r^2 h$

Where r is the radius and h the height.

You're looking for $\frac{dV}{dt}$

By using the multiplication rule, you have :

$\frac{dV}{dt}=\frac{\pi}{3} \left( \frac{dh}{dt} r^2+2h \frac{dr}{dt} r \right)$

From the text, we know that $\frac{dh}{dt}=-4, \ \frac{dr}{dt}=6, \ r=12, \ h=36$

So it should give you :

$\frac{dV}{dt}=\frac{\pi}{3} (-4*144+2*36*6*12)=\dots$

3. I believe your formula for the cone is $\frac{1}{3}pi r^2h$ for the cone. Would you just not take the derivative as you insert your variables. I didn't know there was a Cal IV I thought they only went to III (just an aside to myself )

4. Yeah, listen to Moo, as he understands volume formula much better than myself

5. For the area, it's the sum of the area of the base and the lateral area.

The area of the base is given by $\pi r^2$

The lateral area is given by $\pi r \sqrt{h^2+r^2}$ (I can try to explain it, but later...)

Hence $A=\pi r^2+\pi r \sqrt{h^2+r^2}$

Once again, you're looking for $\frac{dA}{dt}$

Can you do it ?

6. hello,
I am not so sure about the second part (lateral area) and the dA/dt what is the dA/dt? can you explain in details please because I am little confused on the second part Thanks!

7. It means "the derivative of A in respect with t".

8. How do you have take the derivative , I forgot whether I use the chain rule or
other rules, if so how do I do it?

9. Originally Posted by googoogaga
How do you have take the derivative , I forgot whether I use the chain rule or
other rules, if so how do I do it?
Are you needing reminders on how to take a derivative in general or how to take a partial derivative? With partial derivatives you want to take the derivative in the respect to the variable in question. An example would be like

$x^2+y^2+3$ in respect of x would be $2x$

Check out some rules online if your still a bit confused on it

10. I am so sorry for the late thanks I just got so caught up with school work, you know it's time for exams. Thanks again! You clarified it!

11. You clarified it!
You know, this is the most important thing... Sincerely