Optimizing Beam Strength: Finding the Dimensions for Maximum Performance

[Emethyst]

Homework Statement

A rectangular beam is cut out of a cylindrical log with diameter 30cm. Given that the strength of a beam is jointly proportional to its width and the square of its depth, find the dimensions of the beam with maximum strength.

Homework Equations

implicit differentation

The Attempt at a Solution

Frankly, I do not know where to start with this question. I think it's the strength reference that threw me off from this. From what I can tell, S is equal to w+d^2 (where S is strength, w is width and d is depth). Next I need to find another relation to end up with one variable to successfully use the Algorithm for Extreme Values and then find the dimensions. I know I will have to bring the diameter of the log into the problem here somehow, but I am not sure how to do this. If anyone could be of help here it would be great. Thanks in advance.

 

[rootX]

What's depth?
(the length of the rod?)

If it's length, then I think you can get infinite strength (looks somewhat wrong).

It's w.d^2 (not w+d^2) btw as it says "and".

If depth is the base height .. then this is a simple problem. You are given a,b, and a^2+b^2 = diameter^2

 

[Mark44]

The depth is the height of the beam, which will be something less than the diameter of the log from which the beam is cut. The way to visualize this is by drawing a circle that represents a cross section of the tree. Two vertical cuts and two horizontal cuts will produce a beam with a rectangular cross section.

What this problem is saying is that, because gravity works vertically, it's better to have a tall beam than a wide beam (both beams are the same length).

The length of the beam, which would be the length of the section of tree from which the beam is cut, doesn't enter into this problem.

 

[Emethyst]

I have figured that depth would be the height, and that then gives me 2 variables I have to work with. What I am still lost on is what the formulas are for this question; I know one will be the strength formula and the other a relationship used to cancel out one of the variables. I just seem to be stuck on finding the other relationship.

 

[Mark44]

Unless you waste a lot of wood milling the beam, the length of the beam's diagonal will be the radius of the log. That should give you another relationship between width and height.