Minimizing weight of a cylinder using Lagrange multipliers

[mmont012]

Homework Statement

Julia plans to make a cylindrical vase in which the bottom of the vase is 0.3 cm thick and the curved, lateral part of the vase is to be 0.2 cm thick. If the vase needs to have a volume of 1 liter, what should its dimensions be to minimize its weight?

Homework Equations

V=pi*r^2*h
S=2pi*r*h+2pi*r^2

The Attempt at a Solution

I am having trouble with the set up of the problem. I know that the constraint will be the volume, it needs to equal 1 liter.
Thank you in advance to any help that is given.

 

[LCKurtz]

Homework Statement

Julia plans to make a cylindrical vase in which the bottom of the vase is 0.3 cm thick and the curved, lateral part of the vase is to be 0.2 cm thick. If the vase needs to have a volume of 1 liter, what should its dimensions be to minimize its weight?

Homework Equations

V=pi*r^2*h
S=2pi*r*h+2pi*r^2

The Attempt at a Solution

I am having trouble with the set up of the problem. I know that the constraint will be the volume, it needs to equal 1 liter.
Thank you in advance to any help that is given.

Shouldn't your equations have a $.2$ and a $.3$ in them?
What is your objective function (what you are trying to minimize)?
What is your constraint?
Once you have those, write the equation with the LaGrange multiplier $\lambda$ you are going to work with. That should get you started.

 

[mmont012]

The equations that I posted are just the general equations for volume of a cylinder and the surface area of a cylinder. I don't know how to go about setting up the equations to find what I need to. Yes they should have 0.2 and 0.3 in them, but I don't know where to put them. This is the first cylinder optimization problem that I have done.

I said in my original post that the constraint equation will be related to volume since it has to hold 1 liter of water.

If someone can show me how to set up these first two equations that will be super helpful, after that I can do everything else on my own.

 

[LCKurtz]

The equations that I posted are just the general equations for volume of a cylinder and the surface area of a cylinder. I don't know how to go about setting up the equations to find what I need to. Yes they should have 0.2 and 0.3 in them, but I don't know where to put them. This is the first cylinder optimization problem that I have done.

Which of the $.2$ and $.3$ numbers go with the bottom and which with the curved side? Put them in the $S$ equation appropriately.

I said in my original post that the constraint equation will be related to volume since it has to hold 1 liter of water.

So, write down the equation that says the volume = 1.

Then, you didn't answer my question of what you are trying to minimize. The problem says minimize weight. Does that have anything to do with what you have called $S$?

 

[Ray Vickson]

The equations that I posted are just the general equations for volume of a cylinder and the surface area of a cylinder. I don't know how to go about setting up the equations to find what I need to. Yes they should have 0.2 and 0.3 in them, but I don't know where to put them. This is the first cylinder optimization problem that I have done.

I said in my original post that the constraint equation will be related to volume since it has to hold 1 liter of water.

If someone can show me how to set up these first two equations that will be super helpful, after that I can do everything else on my own.

Your equation for $S$ has $2 \pi r^2$ in it, which is the area of the top plus the bottom. Do you really want to make a vase that is closed, like an unopened can of soup?

Anyway, $S$ is not the issue; the issue is the weight of the material used to make the vase. How can you express that in terms of the surface areas you have already calculated?

Please take the time to really try it yourself! If you are having trouble formulating the problem, look in your textbook or course notes for similar examples; if this does not help, do some on-line searches for similarly-titled problems.

 

[mmont012]

All of the examples in my textbook and examples that the teacher worked out in class have dealt with boxes; whether it be minimizing cost of materials or finding the dimensions of a box with and without a lid.

The two equations that I have listed above are just the generic equations that the textbook has for the inside cover. I don't know how to mold these equations into what I need to finish the rest of the problem.

Before posting, I tried a google search for the problem and I found it completely worked out on chegg (which I would have to pay to see) and someone else posted the problem to get help on it, but there wasn't a post in response to it.

Also, the problem is an even problem which means that the solution isn't in the back of the book.

I need to get this homework problem right, because there is no partial credit and it is worth quite a chunk of the grade (meaning if I miss it, I will get a C).

I will try google again since I'm having trouble finding help.

 

[LCKurtz]

You could try answering my questions in post #4.

 

[Ray Vickson]

All of the examples in my textbook and examples that the teacher worked out in class have dealt with boxes; whether it be minimizing cost of materials or finding the dimensions of a box with and without a lid.

The two equations that I have listed above are just the generic equations that the textbook has for the inside cover. I don't know how to mold these equations into what I need to finish the rest of the problem.

Before posting, I tried a google search for the problem and I found it completely worked out on chegg (which I would have to pay to see) and someone else posted the problem to get help on it, but there wasn't a post in response to it.

Also, the problem is an even problem which means that the solution isn't in the back of the book.

I need to get this homework problem right, because there is no partial credit and it is worth quite a chunk of the grade (meaning if I miss it, I will get a C).

I will try google again since I'm having trouble finding help.

I assume you mean that you have already formulated (and maybe solved) problems about rectangular boxes, such as (i) minimizing the cost of the sides and bottom, subject to a constraint on volume; or (ii) maximizing the volume subject to a constraint on the cost of the sides and bottom, and where those costs are related in some way to the areas of the sides and the bottom. If so, you are facing EXACTLY the same problem here; the only difference is that you need different volume and surface area formulas now---but you already wrote these out in post #1!

You really do have all the tools you need; it is just a matter of "thinking outside the box", no pun intended.