Optimization cylindrical can Problem

[Tido611]

Prove that any cylindrical can of volume K cubic units that is to be made using a minimum of material must have the height equal to the diameter.

 

[quantumdude]

Hi,

I've moved your thread to the homework help section. In order to receive help you must show how you started this problem.

Thanks,

Tom

 

[Tido611]

all I've got so far is 2(pi(1/2x)^2)+(x^2pi) and all that explains is the area of any can given the height/diameter. But i don't know how to prove it with calculus.

 

[quantumdude]

Your equation doesn't make any sense to me. For starters it only has one variable, but it should have two.

Let $d$ be the diameter and let $h$ be the height. Can you state expressions for the total volume $V$ and the total surface area $S$ in terms of $d$ and $h$?

If so then you're almost home.

 

[Tido611]

there is no reason to separate d and h from each other because they are the same value d=h=x

and the equation for volume is (pi)r^2x

 

[Hurkyl]

there is no reason to separate d and h from each other because they are the same value d=h=x

No they are not:

(1) Firstly, the diameter of a cylinder is an entirely different thing than the height of a cylinder.

(2) Secondly, nothing in the optimization problem even (directly) suggests that d and h should be equal... it says that the answer should have d and h equal.


If you're still confused by (2), consider this analogy:

You are asked to prove or disprove that the only solution to the equation $x^2 + 3x + 2 = 0$ is $x = 2$. How would you do it?

 

[quantumdude]

there is no reason to separate d and h from each other because they are the same value d=h=x

*thumps head*

You're supposed to show that, not assume it!

and the equation for volume is (pi)r^2x

OK, how about surface area? You're going to need that, too.

 

[nevets93]

ok so the whole point of this problem is to solve everything and then compare d to h in the end

so the surface area being 2pi*r*h + 2pi*r^2, would you find the derivative of this and set it equal to 0?