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, then the are A of the window in terms of x is :
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → $A=5x-\frac{x^2}{2}-\frac{\pi x^2}{8}$
Given
$x+2y+\frac{πx}{2}=10⇒2x+4y+πx=20$
so $y=\frac{20-(π+2)x}{4}$
so area = $xy+\frac{πx^2}{4}$
$A(x)=\frac{x(20-(π+2)x)}{4}+\frac{πx^2}{4×2}$
$A'(x)=\frac{(20-(π+2)x)}{4}-\frac{(π+2)x}{4}+\frac{2πx}{4×2}=0$
$⇒\frac{20-(2π+4-π)x}{4}=0$
$⇒x=\frac{20}{4+x}$
area (x) = $xy+\frac{πx^2}{4}$
$=x(\frac{20-(π+2)x}{4})+\frac{πx^2}{4×2}$
$=5x-\frac{x^2}{2}-\frac{\pi x^2}{8}$