Posted by **Evie** on Saturday, April 19, 2008 at 12:33am.

Show that the rectangle with the largest area that is inscribed within a circle of radius r is a square. Find the dimensions and the area of the inscribed square.

My respect goes to those who know how to tackle this one.

- Calculus -
**Reiny**, Saturday, April 19, 2008 at 7:56am
Let the base of the rectangle be x, let its height be y units.

the diagonal would be the diameter of the circle and it length is 2r.

so x^2 + y^2 = 4r^2

Area of rectangle = xy

= x√(4r^2 - x^2)

d(Area)/dx = ......

= (4r^2 - 2x^2)/√(4r^2 - x^2)

set this equal to zero for a maximum area and solve to get

x = r√2

put this back into x^2 + y^2 = 4r^2

to get y = r√2

so x=y, proving the rectange is a square

- Calculus -
**Evie**, Saturday, April 19, 2008 at 6:02pm
Thank you so much. I appreciate you taking the time to answer my question. = )

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