Integrals

Without thinking too much, I know that a sphere has the highest volume to surface area ratio of any regular solid. So a good first approximation would be a cylinder with a height equal to its diameter.

The constrain is a cylinder for which the ratio Volume/Area is a maximum.

V= π/4* d^2*h A= 2* π*d^2/4+π*d*h if we consider h= m*d we obtain:

α = V/A = d*(2*m)/(1+(2*m)) The function (2*m)/(1+(2*m)) has no extreme it goes from 0 at m=0 to 1 for m=∞. There is no extrema. You can as well check it making the derivative with respect to "m" and set it =0.

You may choose the geometry which seems to you the best.

Thanks. But is there a way to know what width of a wood plate and what type of wood is best for connecting a plate to a wood rod by using calculus. I tried to connect a plate to the end of a small dia wood rod but the wood split as I was screwing thru the plate into the end of the rod.. I thought using the tin can dimension ratio of height vs. dia. might lead to the plate and rod comparison, Far fetched I am sure. What would be the smallest width of a wood plate, knowing its numerical strength factor, that would insure a good connection to the end of a wood rod.

Then, of course, the plate would have to be sawed to enable it to fit snugly inside the roof of the sand mold, leaving the rod protruding thru a hole in the center of the roof of the sand mold.

Another possible calculus use to this whole thing is determining g what is the break thru , eutitic (spelling?) or whatever, point wherein the push rod would work?

Your problem has nothing to do with calculus. It is a simple mechanical problem. When the screw penetrates the rod it has an edge effect and the rod splits since the transverse resistance of wood is a lot less than the one along fibres.

Since the rod is for pushing it could be better to drill first a hole and then screw the plate on. Use screws for wood and only the outer part of thread will enter the wood so the splitting force will be less.

Thanks. I think we are not on the "same page" however. I am talking about the slab of wood splitting, not the rod. I guess the same point could be used for the wood plate. The wood plate would need to be drilled and then use wood screws and only the outer part of thread will enter the wood (?). Edge effect? Simple mechanical problem? How does this explain the thinnest width of the wood slab?

Calculus can be used for obtaining limits, etc. I wanted to know what would be the thinnest slab of wood which would guarantee connection between the slab of wood and rod. I guess transverse resistance of wood, drilling a hole, all come into the picture. Length of screw, tightness, etc.

What about the word eutectic. Point where a liquid becomes a solid. Gas, liquid, solid. In this case: Point where the sand can't be pushed out by push rod. Friction, density, volume, force all come into play. Big problems.

Riley Lohr talks about the problems of pushing the sand out in his paper. Don't worry, I have his letters when he started working on my drawing.

This idea might never work. Oh, well. I haven't given up though but it is not worth it to make too big a deal out of all this.

First, define "wood". Common, commercially available wood species vary dramatically in mechanical properties.

In our encylopedia it says wood is the tough inner portion beneath the bark of trees, shurbs and certain plants.

The manner in which the wood fibers are aligned often results in grain irregularities and distortions Therefore one could rarely get an exact value. But one could give an hypothetical value to the wood slab and this would give an estimated answer I would have no idea what type of wood to use. Examples for types -beech, birch, cherry, chestnudt, elm, mahogany, etc. Here is a good one - I might as well get practice.,

v

Thank you.

Instead of a pail, use a cylinder with a solid cylindrical push block that just barely fits inside. No need to calculate anything!

Eutectic has even less to do with this than calculus.

Just try out a few disks of various thicknesses, and a few push rods of various diameters, say around 5/8 inch.

That sounds good.

Here is a page from my old calculus book (Bacon-Differential and Integral Calculus). This image should be enlarged. I tried Window and clicked zoom) It seems to go in and out and doesn't stay put, it goes back to where it started, a smaller version.

I think this page goes "hand in hand" with the picture of the wood rod and slab of wood along with the sand castle mold and the broken plastic rod & plate. I tried to combine the two but I get a spinning ball instead of a small arrow. I think there are a lot of gliches in this computer business, particularily the password problem and spinning ball in this case.

I can submit examples to enhance this idea that are in the Bacon book

This picture could conceivably start the reasoning of the use of calculus. Using Bacon's book on Differential and Integral Calculus there are examples of a manufacturer making an aluminum cup of fixed volume V of (right circular) cylindrical shape open the top. What proportions will require the least material of uniform thickness? The problem amounts to finding the proportions of a cup of fixed volume and minimum surface area. Another example involves a rectangular box of fixed volume V which is to be twice as long as it is wide. (This one not exactly on the mark for this discussion but shows the use of calculus) It all takes a bit of math review.