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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.

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.

Grade:12

2 Answers

Rinkoo Gupta
askIITians Faculty 81 Points
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
AskIITians Faculty
Arun Kumar IIT Delhi
askIITians Faculty 256 Points
9 years ago
Hi


Thanks & Regards, Arun Kumar, Btech,IIT Delhi, Askiitians Faculty

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