# A large window has the shape of a rectangle surmounted by an equilateral triangle.If the perimeter of the window is 12 metres find the dimensions of the rectangle that will produce the largest area of the window.

Rinkoo Gupta
9 years ago
let the width and length of rectangle are y and x respectively .each sides of the triangle will be x.
so Total area of window S = area of rectangle +area of trianle
=xy+(sqrt3)/4 .x2
and total perimeter 3x+2y=12 =>y=(12-3x)/2
S= x(12-3x)/2 +sqrt3/4.x2
=6x+(sqrt3/4 -3/2) x2
ds/dx=6+2(sqrt3/4 -3/2 ) x
put ds/dx=0
then we get x=12/(6-sqrt3)=4(6+sqrt3)/11
d2s/dx2=2(sqrt3/4 -3/2) <0
so S ios maxima at x=4(6+sqrt3)/11
hence x=4(6+sqrt3)/11 and y=12- 12(6+sqrt3)/11=12(5-sqrt3)/11
Hence the dimensions of the rectangle are 4(6+sqrt3)/11, 12(5-sqrt3)/11
Thanks & Regards
Rinkoo Gupta
Arun Kumar IIT Delhi
$\\b+2a+2a=p \\=>b+4a=12 \\=>ab+{\sqrt3a^2 \over 4}=A \\=>a(12-4a)+{\sqrt3a^2 \over 4}=A \\=>{dA \over da}=12-8a+{\sqrt3a \over 2}=0 \\=>a({\sqrt3 \over 2}-8)=-12 \\=>a={12 \over (8-{\sqrt3 \over 2})}$