How do you find the shape of a rectangle of maximum perimeter that can be inscribed in a circle of radius 5 cm?

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Answer 1 · 2015-04-27T15:59:49

Define the problem

Look at the figure below:

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We will find the point $N(x,y)$
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Perimeter, $P = 4x+4y$

In the question here, $r=5$ , so we have:

$x^2 + y^2 = 25$ , so

$y=sqrt(25-x^2)$

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Therefore, the problem is:

Find $x$ to maximize:

$P = 4x+4sqrt(25-x^2)$ , with $0<= x <= 5$

Solution:
We find critical numbers:

$P' = 4 + (4x)/sqrt(25-x^2)$

$P'=0$ at $x=-sqrt(25-x^2)$ , so

$2x^2 = 25$ , and so:

$x=5/sqrt2 = (5sqrt2) /2$

$P(0)=P(5)=20$

Whe $x = (5sqrt2) /2$ , we also get $y = (5sqrt2) /2$ , so

$P((5sqrt2) /2) = 4((5sqrt2) /2)+4((5sqrt2) /2) = 20sqrt2$

Because $20sqrt2 > 20$ , the maximum value of perimeter occusu whan the rectangle is a square with sides: $(5sqrt2) /2$ ,