Let represent the given perimeter. Let represent the width of the rectangle and let represent the length of the rectangle.
Solve for :
The measure of the diagonal of a rectangle can be found using Pythagoras:
Let represent the measure of the diagonal, and then is the measure of the diagonal as a function of :
An extreme point can be found where the first derivative is equal to zero.
Which is equal to zero when the numerator is equal to zero, namely:
Which is to say when
This extreme point is a minimum if the second derivative is positive at this point.
I'll leave it as an exercise for the student to verify that the second derivative is positive for any positive number P and where .
Since the width is one-fourth of the perimeter, the length must also be one-fourth of the perimeter and the rectangle with the minimum diagonal for a given perimeter is a square with sides that measure one-fourth of the perimeter.
John
My calculator said it, I believe it, that settles it
