Math 22 Optimization Problems

Solving Optimization Problems

Find Maximum Area: Find the length and width of a rectangle that has 80 meters perimeter and a maximum area.

Solution:
Let $l$ be the length of the rectangle in meter.
and $w$ be the width of the rectangle in meter.
Then, the perimeter $P=2(l+w)=80$, so $l+w=40$, then $l=40-w$
Area $A=l.w=(40-w)w=40w-w^{2}$
$A'=40-2w=0$, then $w=20$, so $l=40-w=40-20=20$
Therefore, $l=w=20$

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