Question

Course notes: 1.5 Mass example 2 standard kilogram 
and
Slide: "Proportions of Standard Kilogram"
Calculation of minimum surface area of the standard kg cylinder

My question: why is 2V/r substituted for the area of the cylinder side in the area function A below? 

V =(pi)sq(r)h
A=2(pi)sq(r) +2(pi)rh
   =2(pi)sq(r) +2V /r

This results in a final r= +h/2 instead of r= -h/2 without the substitution
A=2(pi)sq(r) +2(pi)rh
dA/dr =4(pi)r + 2(pi)h=0
r= -h/2
compare to
A=2(pi)sq(r) +2V /r
dA/dr = 4(pi)r - 2V/sq(r)=0
so 
4(pi)r -2 (pi)sq(r)h/sq(r)=0
r = +h/2

is there any ph

Answer 1

It doesnt matter in end. the reason is, to calculate the radius through the volume, which we know, but the height we dont, so there's not much use to know r = h/2. we just know the proportion. So, through the volume we can get both radius and height faster.
You will have to connect the volume with the radius later anyway, it's just gonna take you more time and trouble.

Answer 2

Not 100% sure yet, but I think the volume substitution is necessary. The area needs to be minimized in terms of the single variable r, so the variable h is eliminated by substituting 2V/r where volume V is a constant. Doing it this way gets rid of the unnecessary minus sign.

Answer 3

The radius r cannot be negative. Therefore the substitution is necessary.