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