Just working on an optimization question and was curious as to whether anyone could point out incorrect working, or whether there is a much easier way to solve this problem.

A rectangular beam is to be cut from a cylindrical log with diameter 30 cm. Show

that the beam with greatest cross-sectional area is in fact square. If 4 rectangular

planks are to be made from the 4 offcuts, find the maximum cross-sectional area of

these planks.

So, i have the equation of the circle as:

Equation for cross-sectional area of beam to be cut:

Differentiate

For maximum

When subbed back in to

So the beam with greatest cross sectional is a square. The area is

Now, trying to find the maximum area of the planks made from the cut offs is where i have trouble, the way i went about it was changing the origin of the circle so that the chord that the first rectangle makes is in line with the origin. That is:

The area of the rectangle bounded by the x-axis and the circle is given by:

So substituting the circle equation:

Expanded:

And here is where it gets messy for me:

In cleaning that up, and trying to find the maximum i ended up with

the equation:

From which i got the result:

And substituting to find x i got the result:

All that looks like a horrible mess to me, but the area that results sort of seems reasonable because putting those into the equation:

i get approximately

Using the formula:

I find the area of the segment itself to be approx.

Does my working seem okay on this? I feel like there must be a much simpler way to approach this problem that i am just not thinking of. Sorry about the size of the post and if my working and writing are a bit confusing, this is my first post, and it took me a while to work out how to use the math tags.

Thanks for any help

Regards James