Optimizing Cost for Given Volume of a Cylinder

[toasticles]

if Cost(r)=0.01(2\pir²+v/\pir²(2\pir))+0.015(4\pir+v/\pir²)
v=hpir²
how does one show for the optimal radius/height for any volume i.e. where the cost will be the minimum for that volume?

all my attempts have ended with 0=0

help please? i'll give you cookehz ^^

 

[HallsofIvy]

The first thing I would do is go ahead and replace "v" by \pi r^2 h:
cost(r, h)= 0.01(2\pi r^2+[v/(\pi r^2)](2\pi r)+0.015(4\pi r+v/(\pi r^2)
cost(r, h)= 0.01(2\pi r^2+[(\pi r^2h)/(\pi r^2)](2\pi r)+0.015(4\pi r+(\pi r^2 h)/(\pi r^2)
cost(r, h)= 0.01(2\pi r^2+ h(2\pi r)+ 0.015(4\pi r+ h))

Now differentiate with respect to both r and h:
Treating h as if it were a constant,
cost_r(r, h)= 0.01(4\pi r+ 2\pi h+ 0.015(4\pi))
cost_h(r, h)= 0.01(2\pi r+ 0.015)

To find the optimum height, set those both equal to 0 and solve the two equations for r and h. (Since there is no "h" in the second equation , it is particularly easy to solve it for r. Although it looks to me like there is a sign error- that will give a negative value for r.)

 

[toasticles]

cost(r, h)= 0.01(2\pi r^2+ h(2\pi r)+ 0.015(4\pi r+ h))

i don't think this is right or if it is how did you get it so that the second part(0.15(4pir+h) is also multiplied by 0.01 as the 0.01 and 0.015 "segments" of the equation are separated completely by the addition sign? (i'm only first year calculus) so if it's a law or something please explain :D

 

[lanedance]

if Cost(r)=0.01(2\pir²+v/\pir²(2\pir)+0.015(4\pir+v/\pir²)
v=hpir²

i think its because your original brackets were a little ambiguous - why not use the method but with what you've got?