SOLUTION: A rectangle has a perimeter of 20 meters.3 rectangles were made 1x20, 2x10, and 4x5. How do I find the dimensions of the rectangle with the greatest area. And how do I find the di

Question 1060742: A rectangle has a perimeter of 20 meters.3 rectangles were made 1x20, 2x10, and 4x5. How do I find the dimensions of the rectangle with the greatest area. And how do I find the dimensions of the rectangle with the least area
Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
perimeter is 20, so make length x
width is 10-x, since length + width = half the perimeter
area is x(10-x)=10x-x^2
The derivative of that is 10-2x=0
2x=10
x=5 meters, which is a square 5 meters on a side.
The second derivative is -2. This is differentiable throughout, and the negative second derivative makes it a maximum.
The maximum area of a rectangle with fixed perimeter occurs when the rectangle is a square.
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As the rectangle becomes longer and narrower, the area becomes less
at 1*9 (perimeter 20) the area is 9
at 0.5*9.5 (perimeter also 20), the area is 4.75
at 0.1*9.9, the area is 0.99
at 0.01*9.99, the area is 0.0999.
the dimensions of the rectangle with the least area approach 10 by 0.

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