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.

Question

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

Reiny · Answer

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

Evie · Answer

Thank you so much. I appreciate you taking the time to answer my question. = )