The window of the house is in the form of a rectangle surmounted by a semi-circular position having a perimeter of 10 m as shown in figure.

If x and y represent the length and breadth of rectangular region the whole area of whole window is maximum, then value of x is :

$\frac{20}{4+\pi }$

The correct answer is Option (4) → $\frac{20}{4+\pi }$

Given

$x+2y+\frac{πx}{2}=10⇒2x+4y+πx=20$

so $y=\frac{20-(π+2)x}{4}$

so area = $xy+\frac{πx^2}{4}$

$A(x)=\frac{x(20-(π+2)x)}{4}+\frac{πx^2}{4×2}$

$A'(x)=\frac{(20-(π+2)x)}{4}-\frac{(π+2)x}{4}+\frac{2πx}{4×2}=0$

$⇒\frac{20-(2π+4-π)x}{4}=0$

$⇒x=\frac{20}{4+x}$