Question 1107699
2L + 2W = 400
simplifying:
L + W = 200
L = 200 - W 
Equation for the area:
A = (200 - W) x W = 200W - W^2
So we have a negative quadratic, which means we'll have an upside down parabola where the vertex is the maximum.
Quadratic equation in the format y = ax2 + bx + c :
A = –W^2 + 200W
The vertex of the parabola is the point (h, k) where h = -b/2a:
h = -200/(2 x (-1)) = -200/-2 = 100
To find k, let's plug 100 for W:
k = -100^2 + 200(100) = -10,000 + 20,000 = 10,000
OK, so now we know that h (the maximizing width) is 100, and k (the maximum area) is 10,000. So the shape of our rectangle will be the square:
2L + 2W = 400
2(100) + 2(100) = 400
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NOTE: Of the rectangles (four sided figures with four right angles), the square will always give you the largest area.
And of all geometric figures, the one that will give you the biggest area is the circle. IF your figure was a circle, the area you would get would be:
A = C^2/4Pi where C is the perimeter
A = 400^2/12.56 12,739
Happy learning,
John