Find the are of the largest rectangle that can be inscribed in a semi-circle of radius 3

Answer 1

Draw a figure with the semi-circle on the x axis and the y axis as the bisector of the diameter.

The length of the inscribed rectangle is 2x and the width (height) = sqrt(9-x²)

Area = 2xsqrt(9-x²)

Set the derivative of this equal to 0 and solve for x. I won't even try to write the expression for the derivative in this editor! I get x = (3/2)sqrt(2) for x. You can calculate the area from that.