Question

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.

Answer 1

`2a + b + pi b/2 = 10`
`a = (10 - b - pi b/2)/2`
area = `ab + pi/2*b^2/4`
`= ((10-b-pib/2)b)/2 + (pib^2)/8`
`= 5b^ - b^2/2 - (pi b^2)/4 + (pib^2)/8`
`= 5b - b^2/2 - (pib^2)/8`
`(dA)/(db) = 0`
`=> 5-b- pi/4*b = 0`
`b= 5/(1 + pi/4)`
`(d^2A)/(db^2) = 0-1-pi/4 <0 ` <br> `a = (10-b-pi b/2)/2 = 10/(pi +4)`
area = `(20/(pi+4))(10/(pi+4)) + pi/2*400/(4(pi+4)^2)`
`= (200+50 pi)/(pi+4)^2 = 50/(pi+4) unit^2`
answer