Question

Tough maximization problem!

http://i.stack.imgur.com/Oh2Iw.png

a. Write the area A of the cross as a function of x that maximizes the area.
b. Write the area A of the cross as a function of (theta) and find the value of (theta) that maximizes the area.
c. Show that the critical numbers of parts (a) and (b) yield the maximum area. What is that area? (I think I could do this part.)

Answer 1

This looks ... fun.

Answer 2

Fun indeed, even more fun if you are as ignorant as I am in regards to trigonometry.

Answer 3

have you tried anything yet

Answer 4

Well, looking at those who have attempted the problem before me it seems that they have drawn a right triangle like so..

http://img593.imageshack.us/img593/5237/cross.png

I'm guessing you would set the bottom side to r and then try to solve that triangle?

Answer 5

Though that would be incorrect because, there there is still some space between the cross and the circle.

Answer 6

First of all notice that \(r\) is a given here and it is constant

Answer 7

part a should be easy to work - all you need is to find the other length in terms of \(r\) and \(x\)

Answer 8

For everyone's convenience:
I'm assuming \(r\) is kept constant.

Consider the triangle:

You see that
\[\cos\frac{\theta}{2}=\frac{x}{r}\]
We might need that later...

The missing side to one of the half triangles is found with the Pythagorean theorem:
\[r^2=x^2+?^2~~\Rightarrow~~?=\sqrt{r^2-x^2}\]
which means the sides of the cross closest to the circle's edge is \(2\sqrt{r^2-x^2}\).