Quadratic Function and Diagonals of a Diamond

Date: 6/14/96 at 14:15:44
From: Anonymous
Subject: Quadratic function and diagonals of a diamond

Hello! My question is the following:

   "'The sum of the diagonals of the following lozenge
     is maximum when angle a=30 dgrs.'
                
                         /I\
                        / I \
                       /  I  \
                      <------->
                       \  I  /
                        \ I /  The sides measure sqrt2
                         \I/ 

    Angle a is the angle between the biggest diagonal and the sides."

I tried to answer this question by using a quadratic function, and
by calculating its maximum value [-(b^2 - 4ac)/4a]. Am I right?                        

Date: 6/14/96 at 18:27:43
From: Doctor Anthony
Subject: Re: Quadratic function and diagonals of a diamond

The length of side = L and angle between the diagonal and side = A

The Total of both diagonals is then T = 2L*cos(A) + 2L*sin(A)

To find maximum or minimum value for T we differentiate:

    dT/dA = 2L(-sin(A) + cos(A)) = 0 

             so sin(A) = cos(A)

                tan(A) = 1  and A = 45 degrees

So the figure would be a square.

and this is a maximum since the second differential 
2L (-cos(A) - sin(A)) would be negative at A = 45 degrees.

Then  T = 2L(1/sqrt(2) + 1/sqrt(2))

        = 2L(2/sqrt(2)) = 2L*sqrt(2)

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/