Class 12

Math

Calculus

Application of Derivatives

A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.

$a=210−b−π2b $

area = $ab+2π ⋅4b_{2} $

$=2(10−b−π2b )b +8πb_{2} $

$=5b_{−}2b_{2} −4πb_{2} +8πb_{2} $

$=5b−2b_{2} −8πb_{2} $

$dbdA =0$

$⇒5−b−4π ⋅b=0$

$b=1+4π 5 $

$db_{2}d_{2}A =0−1−4π <0$

$a=210−b−π2b =π+410 $

area = $(π+420 )(π+410 )+2π ⋅4(π+4)_{2}400 $

$=(π+4)_{2}200+50π =π+450 unit_{2}$

answer