Maximize Vol. of Cylinder w/ Conical End: Solution

[jesuslovesu]

[SOLVED] Maximizing an Object

Homework Statement

What proportions will maximize the volume of a projectile in the form of a circular cylinder with one conical end and one flat end, if the surface area is given?

(there is a picture given with r being the radius, l being the length of the cylinder and s being the length of the cone(outside edge))

Homework Equations

The Attempt at a Solution

Well I can come up with:
SA = $$\pi*r^2+ 2\pi*rl + \pi*rs$$
V = $$\pi*r^2l + \pi / 3r^2*s*sin\theta$$

I've tried both using Lagrange multipliers and just using partial derivatives.
but haven't really come up with anything...
(r, l, s)
$$<2\pi rl + 2/3\pi*r*s*sin\theta, \pi r^2 + 0, 1/3\pi*r^2*sin\theta> = \lambda*(<2\pi r + 2\pi l + \pi s, 2\pi r, \pi r>$$
Then with using the 3 equations I was getting r = 2l and l = lamda, which isn't the case, so I think I must either need another equation or my equations are wrong?

 

[arildno]

What is wrong here is that sin(theta) is NOT independent of s and r!
Hence, you cannot treat it as a constant.


Use therefore the relationship: $$s\sin\theta=\sqrt{s^{2}-r^{2}}$$ in your volume formula.

 

[jesuslovesu]

Thanks for your reply, I am able to get a few relationships between s, r, l, but I can't seem to get the answer yet. (ie r = sqrt(5), s = 3)
I can get
$$\sqrt{s^2-r^2} = \frac{2}{3} s$$
$$r = \sqrt{5} / 3 s$$
$$\lambda = r/2$$
Will my lambda need to change throughout the problem? I seem to consistently be getting the same one, but for whatever reason I can't seem to solve the damn thing.

 

[arildno]

Thanks for your reply, I am able to get a few relationships between s, r, l, but I can't seem to get the answer yet. (ie r = sqrt(5), s = 3)
I can get
$$\sqrt{s^2-r^2} = \frac{2}{3} s$$
$$r = \sqrt{5} / 3 s$$
$$\lambda = 2/r$$
Will my lambda need to change throughout the problem? I seem to consistently be getting the same one, but for whatever reason I can't seem to solve the damn thing.

I don't get what you are saying.

You have FOUR equations given:
The three "gradient" equations, along with the requirement that the surface area equals some constant C.

Your optimal values of r, s and l (and lambda) will depend upon that C as a parameter.

 

[jesuslovesu]

My apologies, the book I'm using put the answer in terms of a ratio, I was thinking it was an actual numerical answer.

 

[arildno]

My apologies, the book I'm using put the answer in terms of a ratio, I was thinking it was an actual numerical answer.

Well, a known ratio is an actual numerical answer, it just shows that there are several combinations that maximizes the volume of the object
in question.
It is directly related to the parameter-based approach I'd prefer.