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

Apr 27, 2015

Define the problem

Look at the figure below:

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

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:

$P = 4 x + 4 \sqrt{25 - {x}^{2}}$, with $0 \le x \le 5$

Solution:
We find critical numbers:

$P ' = 4 + \frac{4 x}{\sqrt{25 - {x}^{2}}}$

$P ' = 0$ at $x = - \sqrt{25 - {x}^{2}}$, so

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

$x = \frac{5}{\sqrt{2}} = \frac{5 \sqrt{2}}{2}$

$P \left(0\right) = P \left(5\right) = 20$

Whe $x = \frac{5 \sqrt{2}}{2}$, we also get $y = \frac{5 \sqrt{2}}{2}$, so

$P \left(\frac{5 \sqrt{2}}{2}\right) = 4 \left(\frac{5 \sqrt{2}}{2}\right) + 4 \left(\frac{5 \sqrt{2}}{2}\right) = 20 \sqrt{2}$

Because $20 \sqrt{2} > 20$, the maximum value of perimeter occusu whan the rectangle is a square with sides: $\frac{5 \sqrt{2}}{2}$ ,