Expert: Paul Klarreich - 12/2/2007

Question
I am studying a Chapter on "Differentiation of Transcendental Functions".

My problem is to solve for the largest Area rectangle given a perimeter of 12' using trig fuinctions.

I know of other methods to solve this problem, but I am having a tough time trying to figure this one out using trig functions.

Thank you so much if you point me in the right direction.

Answer
Questioner:   Mark
Category:  Calculus
Private:  No
 
Subject:  Max/Min using trig functions
Question:  I am studying a Chapter on "Differentiation of Transcendental Functions".

My problem is to solve for the largest Area rectangle given a perimeter of 12' using trig functions.

I know of other methods to solve this problem, but I am having a tough time trying to figure this one out using trig functions.

Thank you so much if you point me in the right direction.
----------------------------------------------------
Hi, Mark,

So you didn't like my solution.  I was afraid of that.  OK, here's another way.  Suppose we draw this: [t is your angle theta]
=======================================================
WARNING: USE COURIER FONT TO VIEW THIS
===================================================



+-------+
|      /|
|     / |
|  r /  |
|   /   |
|  /    |
| /     |
|/t     |
+-------+

base is r cos t, height is r sin t.  

Area is r^2 cos t sin t

But the perimeter must be 12, so:

2 r (cos t +  sin t)  = 12

Solve for r:

        6
r = --------------
   cos t + sin t

A = r^2 cos t sin t

    36 cos t sin t
A = -------------------
   (cos t + sin t)^2


    36 cos t sin t
A = ------------------------------------
   cos^2(t) + sin^2(t) + 2 sin t cos t

    36 cos t sin t
A = -------------------
   1 + 2 sin t cos t

Now remember, sin 2t = 2 sin t cos t, so that is:

    18 sin 2t
A = ------------
   1 + sin 2t

Now diff: (use quotient rule, and who cares about the 18?)

          (1 + sin 2t)(2 cos 2t) - (sin 2t)(2 cos 2t)
dA/dt = 18 -------------------------------------------
                   (never mind)^2


          2 cos 2t + 2 sin 2t cos 2t - 2 sin 2t cos 2t
dA/dt = 18 ---------------------------------------------
                   (never mind)^2

           2 cos 2t
dA/dt = 18 ----------------
           (never mind)^2

Now set the top = 0;

cos 2t = 0 when  2t = 90 degrees, or  t = 45 degrees.

You get your max when theta = 45 degrees and the rectangle is a square.