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 `ABCD` be the rectangle of sides `x` and `2y` inscribed in the seimi-circle with centre `O` and radius r.
Let `OA=OB=y` then `AB=2y`
Let `θ` be the angle, then `x=rsinθ,y=rcosθ`
If `A `is the area of rectangle `ABCD`, then

`A=2yx=2rcosθ.rsinθ`
`A=r ^
2 sin2θ`,
differentiating w.r.t `θ`
`therefore (dA)/(d(theta))=r ^
2 cos2θ....(1)`
For max. or min.,`(dA)/(d(theta))=0`
`=>r ^2cos2θ=0`
`=>cos2θ=0=cos pi/2`
`=>theta=pi/4`
Again differentiating equation `(i)` w.r.t` θ`, we get
`(d^2A)/(d theta^2)=-r ^
2 sin2θ=-4r^2 sinpi/2`
`=-4r^2` at `theta=pi/4`which is negative.
hence for maximum area, `θ=pi/4` and `x=rsinθ`
`=rsin (pi)/4=r/sqrt(2)`
`2y=2rcosθ=2rcos pi/4`
`=2rxx1/sqrt2=rsqrt2`
Hence, the dimensions of the rectangle are `r/sqrt2` and `rsqrt2`
and the area`=r/sqrt2 xx rsqrt2 =r^2 `
sq.units