A window is the in the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is of constant length k then its maximum area is:
Answer & explanation
Correct answer: option 3
Let r be the radius of the circle and dimensions of rectangle are 2x, y. according to question 2r + 2y + πr = K
Area = $2t.y+\frac{1}{2}πr^2$ $f(r)=2r(\frac{K-πr-2r}{2})+\frac{1}{2}πr^2$
$f'(r)=K-2πr-4r+πr=0⇒K=(π+4)r⇒r=\frac{K}{π+4}$
$2y=K-(π+2)\frac{K}{π+4}=\frac{2K}{x+4}⇒y=\frac{K}{π+4}$
Maximum area = $2.\frac{K}{π+4}.\frac{K}{π+4}+\frac{1}{2}π\frac{K^2}{(π+4)^2}=\frac{K^2(4+π)}{2(π+4)}=\frac{K^2}{2(π+4)}$