Logical reasoning says the rectangle with the greatest area inscribed in a circle is a square. The diameter of the circle is then the diagonal of the square.
With a diagonal of length 20, the side of the square is 10*sqrt(2); the area is side squared = 200.
To verify that answer, consider the circle with center at the origin, so the equation is x^2+y^2 = 100. A point on the circle has coordinates and .
The dimensions of the rectangle determined by that point are and ; the area is .
The maximum area is when the derivative of the area function is 0.
To simplify the process of finding where the derivative is 0, move all the variables inside the radical:
Set the derivative equal to 0 and solve:
The dimensions of the rectangle with greatest area are
