A rectangle is inscribed in a semi-circle of radius `r` with one of it – InterviewSolution

1.	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 rectangle so that the area is maximum. Find the area also.
Answer» `A(theta)=widthxxheight` `=(2rcostheta)xxrsintheta` `=2r^2sinthetacostheta` `=r2xx(2sinthetacostheta)` `=sin2theta` `A(theta)=x^2xxsin(2theta)` `sin(2theta)=1` `2theta=pi/4` `theta=pi/4` Maximum area = `r^2xx1=r^2` `width=2xcostheta=2rcospi/4` `=2rxx1/sqrt2` `=sqrt2r` `height =rsintheta=rxxsin(pi/4)` `=r/sqrt2`

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