Daily Archives: August 6, 2008

A mind boggling maximization problem!

So I encountered what I think is a horribly wonderful (and did I mention horrible?) problem. It took me forever to solve, because I kept getting different solutions and I didn’t know which was right and which was wrong and where my errors might have come from. But in the end, I think I finally got it right. The error was a silly sign error which made everything nice.

Here’s the problem [1]:

A length of sheet metal 27 inches wide is to be made into a water trough by bending up two sides as shown in the accompanying figure. Find $x$ and $\theta$ so that the trapezoid shape cross section has a maximum area.

I don’t want to tell you what I got for the answer, in case you want to try it out for yourself, but I will tell you this: I was able to get a maximum area of $\frac{243\sqrt{3}}{4}$ square inches.

(And if you are banging your head against a wall, and need a solution for peace of mind, shout out in the comments. I’d be happy to type out my solution. I just don’t want to spend the time if no one cares.)

Update: I had $\frac{1453}{8}(\sqrt{2}-1)\sqrt{2\sqrt{2}-1}$ posted as the answer, but again, I noticed a second silly sign error. I have never made so many algebraic errors on an problem as I did with this one.