Optimization: Beam of maximum strength

[HappMatt]

first off here is the problem
:18. A rectangular beam is cut from a cylindrical log of radius 30 cm. The strength of a beam of width w and height h is proportional to . Find the width and height of the beam of maximum strength.
so I have a diagram of it but I am having troubles setting up the problem. But i just can't figure out how to set it up. I think that one of the equations i need would be (h*w)/(wh^2) not sure if that's right and then for the sides i figure you have to use the pythagorean identity to get h=(30^2-w^2)^(1/2) from here I am lost and have obviously given up hope of solving it on my own. P.S. i absoulutly hate this stupid huges-huallete/wiley single variable calc book, the book suck, or mybee i do.


- wel I am still staring at the problem and this time i came up with something new. I though mybe i was reading the proportional part wrong and that mybee (w*h^2) is the proportion and that all i need it the deriv of that maximized at 0 then plug in h=(30^2-w^2)^(1/2) to figure out where the height is maximized. i also added some work so you all don't just think I am a lazy bastard.

 

[lippyka]

The strength of the beam is proportional to w*h^2. Taking the derivative of this with respect to width (w) and setting it to zero, we have 2wh=0, which implies w=0. Plugging this into the equation for height, we get h=(30^2-0^2)^(1/2)=30 cm. Therefore, the width and height of the beam of maximum strength are 0 cm and 30 cm respectively.