# Maximizing the volume of a cylinder

• CrosisBH
In summary, the surface area is fixed, so that'll be the constraint. Maximizing volume, we need a functional to represent Volume. This was tricky, but my best guess for it isV = \int dV = \int_{0}^{H} \pi R^2 dH That way it can be represented by a single integral. f = R^2 Plugging the stuff into the Euler-Lagrange equation and simplifying I get (I can show my work here if it's needed)(2\lambda + 1)R + \lambda H = 0 I'm just stuck here. I don't know how to proceed.
CrosisBH
Homework Statement
Find the ratio of R (radius) to H(height) that will maximize the volume of a right circular cylinder for a fixed total surface area.
Relevant Equations
$$\frac{\partial f}{\partial q_k} - \frac{d}{dx} \frac{\partial f}{\partial q_k '} + \sum \lambda_k (x) \frac{\partial g_k}{\partial q_k} = 0$$

$$S = 2\pi RH + 2\pi R^2$$
$$V = \pi R^2 H$$
Note this is in our Lagrangian Mechanics section of Classical Mechanics, so I assume he wants us to use Calculus of Variations to solve it.
The surface area is fixed, so that'll be the constraint. Maximizing volume, we need a functional to represent Volume. This was tricky, but my best guess for it is

$$V = \int dV = \int_{0}^{H} \pi R^2 dH$$

That way it can be represented by a single integral.
$$f = R^2$$
Plugging the stuff into the Euler-Lagrange equation and simplifying I get (I can show my work here if it's needed)

$$(2\lambda + 1)R + \lambda H = 0$$

I'm just stuck here. I don't know how to proceed. Calculus of Variations is still very new to me. Thank you!

CrosisBH said:
I can show my work here if it's needed
Yes it's needed ... I'd like to see what you use as Lagrangian and what you use as ##q_k## and ##g_k##

 You seem to think that ##{\mathcal L} = f = R^2 ## but that is not correct
Furthermore, the number of constraints is not equal to the number of variables, so ##g_k## and ##\lambda_k## looks strange: The sum is not over ##k## but over the constraints :##\displaystyle{
\sum_i \lambda_i \frac{\partial g_i}{\partial q_k}}##

The ##\lambda_i## are numbers, not functions.

- - - - - - - - - - - - - - -

[edit2] looked at your profile and want to step back somewhat:

##f## is the function you want to maximize, so here ## f(r,h) = V = 2\pi r^2 h ##, a function of 2 variables. At an extreme you expect ##{\partial f\over \partial h}= {\partial f\over \partial r} = 0##.
Two equations in two unknowns that are not independent.

Instead of eliminating one of the variables, you use the method of Lagrange multipliers and solve
##{\partial {\mathcal L} \over \partial h}= {\partial {\mathcal L} \over \partial r} = 0## with ##{\mathcal L} = f - \lambda g##.

##g(r,h) = 0## is the constraint ##S = 2\pi r^2 + 2\pi rh \quad \Rightarrow \quad g(r,h) = 2\pi r^2 + 2\pi rh - S##.

Now we have three equations (the third one is ##g=0##) with three unknowns (##r,h,\lambda##) and we can treat ##r## and ##h## as if they were independent.

--

Last edited:

## 1. What is the formula for calculating the volume of a cylinder?

The formula for calculating the volume of a cylinder is V = πr^2h, where V is the volume, π is the mathematical constant pi, r is the radius of the base, and h is the height of the cylinder.

## 2. How do I maximize the volume of a cylinder?

To maximize the volume of a cylinder, you need to increase either the radius or the height of the cylinder. However, there is a limit to how much you can increase these dimensions, as the base of the cylinder cannot exceed the surface area of the cylinder's lateral face.

## 3. What is the relationship between the radius and height of a cylinder in terms of maximizing volume?

The radius and height of a cylinder have an inverse relationship when it comes to maximizing volume. This means that as one dimension increases, the other must decrease in order to maintain a constant surface area and maximize the volume.

## 4. Is there a maximum volume that a cylinder can have?

Yes, there is a maximum volume that a cylinder can have. This is determined by the dimensions of the cylinder's base and lateral face, as well as the mathematical constant pi. The maximum volume occurs when the cylinder is a perfect cylinder, meaning the base and lateral face have the same dimensions.

## 5. How can I use the concept of maximizing volume of a cylinder in real life?

The concept of maximizing volume of a cylinder is used in many real-life applications, such as designing storage containers, water tanks, and even rockets. By maximizing the volume of these objects, we can efficiently use space and resources while still achieving our desired volume.

Replies
9
Views
3K
Replies
10
Views
1K
Replies
4
Views
1K
Replies
10
Views
2K
Replies
8
Views
3K
Replies
1
Views
789
Replies
7
Views
1K
Replies
3
Views
2K
Replies
6
Views
1K
Replies
18
Views
2K