Question

I'm trying to solve the first worked example (the one about the kilogram), and I have the formula for the cilinder's surface A = 2(pi)r^2 + 2(pi)rh. If I differentiate that formula as is, I get dA/dr = 2h(pi)+4(pi)r. When I substitute V I get dA/dr = 2V*r^-2 + 4(pi)r.

In the pdf they substitute V before doing the differentiation, thus getting the formula dA/dr = -2V*r^-2 + 4(pi)r.

Why does substituting V before or after the differentiation yield different results?

Answer 1

Substituting V before differentiation means you will differentiate it (V) after substitution.
Substituting V after differentiation means that you get something in terms of V after differentiation, and this V will not be differentiate after substitution.
So, the results are not (may not be) the same.

Answer 2

@ Callisto- But in the Pdf where the V is taken before consideration, and then differentiated it appears the volume (V) is CONSTANT. Now, since you're differentiating wrt radius, and the radius(r) is not constant, it means the height (h) should also CHANGE to keep the volume constant.
Whereas in the 1st case he seems to have taken 'h' constant (wrongly). I'm not too sure on my 'hypothesis' here. Come on, I'm still in high school!
:P

@robertorrw- Check the problem. Maybe, they have stated that the volume is constant.

Answer 3

Would you mind attaching the pdf here? I am sorry that I haven't read it before I answered this question.

Answer 4

Here's the pdf, on page 7. Seems like V is constant, but why can't it also be a function of the radius?

Answer 5

Look, the volume is definitely constant. Here's why: the mass (m) is fixed at 1kg, the density is also fixed at 21.56 g/cc. So , the volume is m/density= some constant (V).

Now, that we have established that the Volume is constant, let's move on to the other part of your question:
V= (pi)*r^2*h.... now, as I mentioned that to keep the volume constant BOTH the radius and height vary in such an order so as to compensate each other's change. If r increases, h decreases and vice versa. <<NB: BOTH have to change, as otherwise (pi)r^2*h can never be constant. (Eg- if h is constant, but r increases, (pi)r^2*h, as it is directly proportional. So the overall volume shall increase. Discrepancy! )

So, differentiating the area ,(A= 2(pi)*r^2 +2(pi)rh )
dA/dr= 4(pi)*r+[2(pi)h+2(pi)r*dh/dr] << (Note carefully: Since both are variables you have to differentiate according to the rule: d(u.v)/dx= v.du/dx+u.dv/dx)

But, do you know the result of dh/dr? No!
So you need to replace A= 2(pi)r^2+2V/r. :)

Now, vote my 1st answer as the best, 'cause I guessed the volume was constant without even looking! I'm awesomeeeeeee! xD

FYI: Had you differenciated like dA/dh, instead of dA/dr, you would have had to make the necessary substitutions so that the equation had only h as it's function and V as the constant.

Answer 6

Got it, thank you.

Answer 7

robertorrw tanks for your pdf