A window of perimeter `P` (including the base of the arch) is in the form of a rectangle surrounded by a semi-circle. The semi-circular portion is fitted with the colored glass while the rectangular part is fitted with the clear glass that transmits three times as much light per square meter as the colored glass does. What is the ratio for the sides of the rectangle so that the window transmits the maximum light?

Explanation

Let x and y meters be the lengths of the sides of rectangle ABCD and let there be semi circle on side CD opf lengthx.

Therefore perimeter of the window (including the base of the arch)=perimeter of the rectangle +perimeter of the semi circel
`=2x+2y+(1)/(2)(2pix//2)`
`=2x+2y+(1)/(2)pix=c` (constant)
`therefore y=(1)/(2)(c-2x-(1))/(2)pi x`
Let k be the light per square meter transmitted by oclored glass so that transmitted by clear glass will be 3k per square meter Hence the total light transmitted by the window is given by
A=(area of colored gloass)k+(Area of clear galss)3k
`=(1)/(2)pi(x//2)^(2)k+xy(3k)`
`=(1)/(8)pikx^(2)+3kx(1)/(2)(c-2x-(1)/(2)pix)`
`=(1)/(8)k(-5pix^(2)-24x^(2)+12cx)`
`therefore (dA)/(dx)=(1)/(8)k(-10pix-48x+12c)`
and `(d^(2)A)/(dx^(2)=(1)/(4)(5pi+24)k`=-ve
For maximum or minimum of `A,dA//dx=0`
or `x=6c//(5pi+24)`
Therefore from `(1),yf =(pi+6)c//(5pi+24)`
Since `(d^(2)A)/(dx^(2))` is ve, A has maxima
Hence ratio `x//y=6(pi+6)`