Question 985500: Find three rectangles which have a perimeter equal to 100cm and an area greater than 200square cm
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
The largest area rectangle for a given perimeter is a square with sides that measure one-fourth of the perimeter. So a square with a perimeter of 100 has an area of 25 squared, or 625, and that is most certainly larger than 200.
In order to get closer to an area of 200, consider that the sum of the length and the width of a rectangle must be equal to one-half of the perimeter. So for a perimeter of 100, the sum of the length and the width must be 50.
So:
And since the area is the length times the width:
\ =\ -w^2\ +\ 50\)
And you are concerned with those values of length and width such that the area is greater than 200.
If we make this into a quadratic equation and solve for  , we get
roughly  or 
So for one of your rectangles, we already have  , giving us  and 
For the next one, choose any number you like between 4.4 and 45.6 and call that the length. Subtract your chosen number from 50 and call that the width. The perimeter will, perforce, be 100, and you can multiply the length times the width to get the area and verify that it is indeed larger than 200.
For the third one, repeat the last step with a different number in the specified interval.
John
My calculator said it, I believe it, that settles it
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