Question

What is the largest size of rectangle that can be inscribed in a semicircle of radius one unit, so that two vertices lie on the diameter.

Answer 1

${\left(\frac{x}{2}\right)}^2+y^2=1\Rightarrow x^2+4y^2=4$ .....(i)

Max Size $=$ Max Area

$A=xy$ $=2y\sqrt{1-y^2}$

$\frac{dA}{dy}=\frac{2y(-2y)}{2{\sqrt{1-y}}^2}+2\frac{(1-y^2)}{\sqrt{1-y^2}}=0$

$\Rightarrow -2y^2+2(1-y^2)=0$

$y=\frac{1}{\sqrt{2}}$

$x=\sqrt{2}$