Question

find the rectangle of maximum perimeter inscribed in a given circle

Answer 1

see the pic, here is two solutions for this question..

My second solution is by using algebraic expressions in the pic..

for me second solution is easy..

Answer 2

There is a circle of given radius say r.

A rectnalge is inscribed in it.  The rectangle has each angle 90 degrees. Hence the diagonals of the rectangle would be the diameter of the circle by central angle subtended theorem for circles.

Hence rectangle has diagonal as 2r.

Let sides be l and w

Perimeter = 2l+2w

Also from Pythagorean theorem for right angles

$l^2+w^2 = 4r^2</p><p>Or w^2 = 4r^2-l^2$

Our purpose is to make perimeter the largest.

P (l) = 2l+2w  =$2l+2\sqrt{4r^2-l^2}$

Use derivative test.

P'(l) = $2+2(\frac{-2l}{\sqrt{4r^2-l^2} } )$

Equate P'(l) to 0

We get

$(\frac{2l}{\sqrt{4r^2-l^2} } )=1</p><p>4l^2 = 4r^2-l^2</p><p>Or l = \frac{2r}{\sqrt{5} } </p> <p>w= [tex]\frac{12r}{\sqrt{5} }$

The rectangle will have length and width as found above and perimeter

= \frac{12r}{\sqrt{5} }