A closed right cylindrical tank is to have a capacity of 128pi cu.m. Find the dimensions of the tank that will require the least amount of material in making it. (Solution must be with Differential Calculus)

Question

ybarrap · Answer

Using the method of Lagrange multipliers, I get
$$h=\frac{ 8 }{ \pi^{1/3} },r=\frac{ 4 }{ \pi^{1/3} }$$
So basically, a square-looking cylinder, with h=diameter=2r, which is what you might expect.

anonymous · Answer

Thank you! :) :) Can i have the solution please :)

ybarrap · Answer

Do yo know the method of LM?

anonymous · Answer

No.

ybarrap · Answer

http://en.wikipedia.org/wiki/Lagrange_multiplier, read this and then let me know what is f and g?

ybarrap · Answer

You have two unknowns, h and r. Map f and g to surface area and volume, respectively.

ybarrap · Answer

Look at the 1st example in the site, it'll make things clear

ybarrap · Answer

f is what you want to optimize, g is the constraint

ybarrap · Answer

you'll just need to figure out 3 partial derivatives, set to zero and solve for h and r. That's it.

anonymous · Answer

So what are those?

ybarrap · Answer

you need to tell me first what is f and g, then setup your equation like this, but using surface area and volume instead of f and g:

ybarrap · Answer

r=x and h=y, instead -- everything else is just method

ybarrap · Answer

You'll have something like (but with YOUR variables):

ybarrap · Answer

Good luck

anonymous · Answer

Ok thank you again! God Bless you

anonymous · Answer

uh oh, the answer does not match with the volume, it will only result to 128 but the volume is 128pi

ybarrap · Answer

oh, I didn't include the pi factor, but the technique is the same. If you can solve the differentials, you can make this correction

ybarrap · Answer

If you just want an easy answer, knowing that 2r=h, put into formla for V an and solve. Same deal with the other problem. Generally, symmetry of the problem gives you a clue for optimization problems. But I recommend you try to justify with Lagrange

ybarrap · Answer

I haven't see you try yet, sorry. But you can't learn by just getting the right answer.

anonymous · Answer

finally i got the right answer  for the second problem i gave to you. but i cannot understand the first one