Question 1014083: Word Problem: What must be the dimensions of a rectangular lot whose perimeter is 400 ft, if the area of the lot is to be a maximum? Answer by ikleyn(52778) (Show Source):
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Word Problem: What must be the dimensions of a rectangular lot whose perimeter is 400 ft, if the area of the lot is to be a maximum?
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Answer. A square with the side measure 100 ft.
Proof
Since the perimeter of the rectangle is 400 ft, the sum of two adjacent/consecutive sides is half of it, i.e. 200 ft.
Let x be the measure of one side. Then the measure of the other side is 200-x.
The area is x*(200-x) = .
This function is a parabola opened downwards.
Complete the square:
= = .
Now one can see that the parabola has the maximum equal to 10000 at x = 100.
It is what has to be proved.
