Expert: Paul Klarreich - 9/28/2005

Question
Hi, i need help on a question.

A wire 30 cm. long is to be cut into two pieces. One piece is bent into a circle and the other into a square. How long should each piece be so that the sum of the areas of the shapes are maximized?

I got up to the part using the formulas and substitution where
A= (x^2/4pi)+((30-x)^2/16)

I know next (at least i think), you're supposed to find the derivative and set it to 0 to find critical values to test, but i don't know how to get the derivative.
If you can help, I would appreciate it.
Thanks!

                              -Charlene


Answer
Hi, Charlene,
I'm not sure how you set it up (there could be more than one way) but to differentiate:
   x^2   (30 - x)^2
A = --- + ----------
   4pi      16

All you need to do to preserve your sanity is to remember that those denominators are constants.  You just have to differentiate:  (I am assuming you have learned certain basic derivative rules, including the rule for x^n and the Power Rule.)

(1)  x^2, and the derivative is 2x.

(2) (30 - x)^2, which is a power of a function, (30-x).

Its derivative is 2(-1)(30 - x) = -2(30 - x)

So the derivative is:

dA     2x    -2(30 - x)    x    30 - x
--  = ---- + ---------- = --- - -------
dx    4pi        16       2pi      8

Now you can set that to zero and solve.