Practicing Success
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 : |
$A=x-\frac{\pi x^3}{8}-\frac{x^2}{2}$ $A=5x-\frac{x^2}{2}-\frac{\pi x^2}{8}$ $A=5x-\frac{x^2}{2}-\frac{3x^2}{8}$ $A=5x+\frac{x^2}{2}+\frac{\pi x^2}{8}$ |
$A=5x-\frac{x^2}{2}-\frac{\pi x^2}{8}$ |
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) = $xy+\frac{πx^2}{4}$ $=x(\frac{20-(π+2)x}{4})+\frac{πx^2}{4×2}$ $=5x-\frac{x^2}{2}-\frac{\pi x^2}{8}$ |