# Maximum area for inscribed cylinder

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1. May 12, 2017

### Elias Waranoi

1. The problem statement, all variables and given/known data
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 rc and height of cylinder is hc.

2. Relevant equations
A = 2πrchc + 2πrc2

3. The attempt at a solution
r = h ∴ hc = r - rc
A = 2πrc(r - rc) + 2πrc2

To get the maximum of this area I will find the radius rc when the growth of the area is zero.
dA/drc = 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.

2. May 13, 2017

### BvU

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

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

Last edited: May 13, 2017
3. May 13, 2017

### Ray Vickson

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.)

4. May 13, 2017

### Elias Waranoi

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/drc = 0 = 2πr. If for example my cone had the radius r of 1 meter than my derivation would give dA/drc = 2πr = 6.28 = 0 which doesn't make sense. But dA/drc = 0 is just a statement I made and if dA/drc ≠ 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!

5. May 13, 2017

### Ray Vickson

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.

Last edited: May 13, 2017