SOLUTION: why does a square 45x45 have more sqaure footage than a rectangle 30x60. both could be made from the same two 90' pieces

Question 399513: why does a square 45x45 have more sqaure footage than a rectangle 30x60. both could be made from the same two 90' pieces
Found 2 solutions by josmiceli, Alan3354:
Answer by josmiceli(19441) (Show Source): You can put this solution on YOUR website!
You have 180' to go around it, but picture this:
Suppose the rectangle is 1 x 89. The pieces
would be , but the area
enclosed would be ft2
But the 45 x 45 square has area =
That's a difference of ft2.
I rest my case.
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
The simplest answer is that 45*45 is more than 30*60.
For a given perimeter (180 would make a 45 by 45, not 90) the max area of a rectangle is a square.
------------
For a perimeter of 180
P = 2W + 2L = 180
w + L = 90
Area = W*L = W*(90 - W)
To find the max area as a function of W:
f(W) = w(90 - W) = 90W - W^2
90W - W^2 - area = 0
-W^2 + 90W - area = 0
This is a parabola that is open downward.
The line of symmetry is W = -b/2a = -90/-2 = 45
The max is on the line of symmetry:
f(45) = 90*45 - 45^2 = 4050 - 2025
Max area = 2025 sq units
W = 45
L = 90-45 = 45
------------
Any other pair that add to 90 will have a smaller product.
It will be (45 + x)*(45 - x) = 2025 - x^2
For any x > 0, it will be less than 2025.
--------------------
Note: 2025 is the max area of a rectangle with a perimeter of 180.
A circle will have the max area of all figures.

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