As I understand your question, you will want the radius of the circle (top and bottom) to be half the height of the square so that the circle is perfectly encased in the square, thus wasting the lowest amount of material.
The ratio is therefore
I would appreciate help on this question:In constructing a cylindrical can of given volume, nothing is wasted in making the side of the can. But the top and bottom are cut from square sheets and the remainder are wasted. Find the ratio of h to r so that the material used is a minimum.
I've tried several methods but i can't seem to find a practical solution. Please help. Thanks
As I understand your question, you will want the radius of the circle (top and bottom) to be half the height of the square so that the circle is perfectly encased in the square, thus wasting the lowest amount of material.
The ratio is therefore
Nono. If we look at his cylinder from the top down, it will look like this (the material being the square): http://s23.postimg.org/3wkg8ho23/cylinder.png
h in my equation represents the height of the square, not the cylinder.
Obviously this is not calculus as the post's section suggests, but this is how I understand OP's problem.
Here is my understanding of the problem:
The sides are cut without waste, so an area of is used.
The top and bottom are circular, and are cut out of squares, so instead of an area of times 2 for top and bottom, an area of (times 2) is used.
The cylinder has volume .
So the total area is . Substituting gives .
The volume V is fixed - we want to minimize A by varying r. So , .
The problem asked for the ratio of h to r: .
- Hollywood