# Thread: Steel Drum

1. ## Steel Drum

Hello! I am having problems in solving this problem. I don't know where to star but I have some formulas that I started with and I don't know what to do from there.

Your task is to build a steel drum (right circular cylinder) of fixed volume. This time the consideration of waste material is added, but the material cost is still the same for the top and the sides (same gage and same cost) the tops and the bottoms will be cut from sheet metal from squares of length 2r. Use calculus to determine the amount of metal used is minimized when h/r=8/π.

I have just the formulas:
--Area of a cirlce
--Volume of a cylinder
--Area of a square

I will appreciate your help

2. ## Re: Steel Drum

With the radius r and the height h, how much material is used?
Using the formula for the volume (and the fixed volume V), can you find a relation between r and h?
Can you express the used material as function of a single variable (r or h) only?
Do you know how to find the minimum of this function?

These steps are not specific for your problem here, they can be used for all problems of this type.

3. ## Re: Steel Drum

I presume the barrel can be made just by bending a rectangle of width h and length $\displaystyle 2\pi r$, the circumference of the circle. Now, since we have to pay for waste as well as the material used, from "sheet metal from squares of length 2r" we have to pay for the full $\displaystyle 2(2r)^2$. Now what is the total material used for both barrel and ends in terms of h and r? Use the formula for volume to reduce that to one variable.

Because the problem asks for a relation between r and h, rather than explicit values, you might try the "Laplace multiplier" method. It tends to give relations first which could then be solved for explicity values. Do you know that method?

However, doing it both ways, I get the same answer but NOT "$\displaystyle 8/\pi$"!