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Numerade Educator

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Problem 20 Medium Difficulty

(a) Show that of all the rectangles with a given area, the one with smallest perimeter is a square.
(b) Show that of all the rectangles with a given area, the one with greatest area is a square.

Answer

a) Smallest perimeter makes a square
b) Greatest area is a square.

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Video Transcript

we know that if l w is A than P is to l plus two w meaning that two outposts to a over l is gonna be p which gives us p prime. In other words, the derivative is gonna be to minus too, huh? Overall squared. So that's equal to zero When we get our equals square root of a this means w a squirt of it w and our both equivalent discord of Ed, which indicates that because the with equals the length, the rectangle, the smallest perimeter is indeed the shape of a square. Now moving on to part B, we know we can rearrange. You are link terms with equals A to write it in terms of the west length plus with equals perimeter over to their fur with his perimeter over to minus length. They're for the area is length times permit over to minus length squared. Therefore, a prime is p over to Ryan's to Elsa this equal to zero. So for out on me, get out. Is Pete over four Now do the same thing for w we get w is also pee over, for they are equivalent. Therefore, l is equivalent to W. Therefore, we know that because length equals with of our tingle with the maximum area is a square