A rectangle is inscribed in a semi-circle of radius `r` with one of its sides on diameter of semi-circle. Find the dimensions of the rectangle so that its area is maximum. Find also the area.

Explanation

Let the length be `x ` and breadth be `y `.

`{x^2}/4+y^2=r^2`

`impliesx^2=4(r^2-y^2)`

Area of the rectangle`=A=xy`

`implies A^2=x^2y^2=4y^2(r^2-y^2)`

`{dA^2}/{dy}=8yr^2-16y^3`

Put `{dA^2}/{dy}=0`

Thus `8yr^2-16y^3=0 `

...