Maximum area for inscribed cylinder

[Elias Waranoi]

Homework Statement

Inscribe in a given cone, the height h of which is equal to the radius r of the base, a cylinder (c) whose total area is a maximum. Radius of cylinder is r_c and height of cylinder is h_c.

Homework Equations

A = 2πr_ch_c + 2πr_c²

The Attempt at a Solution

r = h ∴ h_c = r - r_c
A = 2πr_c(r - r_c) + 2πr_c²

To get the maximum of this area I will find the radius r_c when the growth of the area is zero.
dA/dr_c = 0 = 2πr

What does this result mean? I don't understand how 0 = 2πr makes sense as a result from a derivation. What kind of information does this result give me geometrically? How can I know that there is no maximum area to the cylinder as my answer sheet tells me.

 

[BvU]

Check your result for dA/dr_c

[edit] oops, I checked my result and now we agree. Looks as if the area does not depend on r_c -- as I could have read in your post

[edit] oops² Thanks Ray for putting me right -- dA/dr_c does not depend on r_c but the area itself of course does. It just keeps increasing until r_c hits r and then you have a pancake, not a cylinder any more. Tricky exercise if you are doing it before coffee

 

[Ray Vickson]

Homework Statement

Inscribe in a given cone, the height h of which is equal to the radius r of the base, a cylinder (c) whose total area is a maximum. Radius of cylinder is r_c and height of cylinder is h_c.

Homework Equations

A = 2πr_ch_c + 2πr_c²

The Attempt at a Solution

r = h ∴ h_c = r - r_c
A = 2πr_c(r - r_c) + 2πr_c²

To get the maximum of this area I will find the radius r_c when the growth of the area is zero.
dA/dr_c = 0 = 2πr

What does this result mean? I don't understand how 0 = 2πr makes sense as a result from a derivation. What kind of information does this result give me geometrically? How can I know that there is no maximum area to the cylinder as my answer sheet tells me.

Your problem is constrained: $\max f = r_c h_c + r_c^2$ subject to $r_c+h_c = r$. The objective is $A/(2 \pi)$ and the constraint arises because the cylinder is inscribed in a cone of base-radius and height $r$. You can substitute $h_c = r - r_c$ into $f$ to get a one-dimensional problem $\max f_1 = r_c^2 + r_c(r-r_c) = r r_c,$ subject to $0 \leq r_c \leq r$. What is the solution to this last problem? (Remember that $r$ is a constant, so $r_c$ is the only variable.)

 

[Elias Waranoi]

Your problem is constrained: maxf=rchc+r2cmaxf=rchc+rc2\max f = r_c h_c + r_c^2 subject to rc+hc=rrc+hc=rr_c+h_c = r. The objective is A/(2π)A/(2π)A/(2 \pi) and the constraint arises because the cylinder is inscribed in a cone of base-radius and height rrr. You can substitute hc=r−rchc=r−rch_c = r - r_c into fff to get a one-dimensional problem maxf1=r2c+rc(r−rc)=rrc,maxf1=rc2+rc(r−rc)=rrc,\max f_1 = r_c^2 + r_c(r-r_c) = r r_c, subject to 0≤rc≤r0≤rc≤r0 \leq r_c \leq r. What is the solution to this last problem? (Remember that rrr is a constant, so rcrcr_c is the only variable.)

Sorry, bad title. What I'm looking for is the maximum in the curve of the growth of area for a cylinder inscribed in a cone. Not the maximum area. What I'm hung up on the result of my derivation dA/dr_c = 0 = 2πr. If for example my cone had the radius r of 1 meter than my derivation would give dA/dr_c = 2πr = 6.28 = 0 which doesn't make sense. But dA/dr_c = 0 is just a statement I made and if dA/dr_c ≠ 0 then that means my statement is false and there is no maximum or minimum in the growth of area for the cylinder. Seems like I figured it out while writing me reply... Sorry for wasting your time and thank you all for trying to help me!

 

[Ray Vickson]

Sorry, bad title. What I'm looking for is the maximum in the curve of the growth of area for a cylinder inscribed in a cone. Not the maximum area. What I'm hung up on the result of my derivation dA/dr_c = 0 = 2πr. If for example my cone had the radius r of 1 meter than my derivation would give dA/dr_c = 2πr = 6.28 = 0 which doesn't make sense. But dA/dr_c = 0 is just a statement I made and if dA/dr_c ≠ 0 then that means my statement is false and there is no maximum or minimum in the growth of area for the cylinder. Seems like I figured it out while writing me reply... Sorry for wasting your time and thank you all for trying to help me!

Setting the derivative to zero is a mistake: you have a constrained problem! That is, your originally-described problem is
$$\begin{array}{l}\max A(r_c) = 2 \pi r_c(r-r_c) + 2 \pi r_c^2 = k r_c, \\
\text{subject to} \;\;0 \leq r_c \leq r.
\end{array}$$
Here, $k = 2 \pi r$ is a positive constant.

Your newly-described problem is to maximize the constant $k$ on the set $0 \leq r_c \leq r$, which makes no sense: the area just keeps growing at a steady rate, so every point is a maximum growth rate point.

Setting a derivative to zero is what you do when you are looking for an interior-point max or min of a function, but not when you are on the boundary of an inequality constraint set. You would never, ever solve $\max/ \min f(x) = 2x, \: 0 \leq x \leq 1$ by setting the derivative of $2x$ to zero.