Grade 12Differential Calculus

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.

taniska

12 Years agoGrade 12

2 Answers

Rinkoo Gupta

12 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

12 Years ago

$\\b+2a+2a=p \\=>b+4a=12 \\=>ab+{\sqrt3a^2 \over 4}=A \\=>a(12-4a)+{\sqrt3a^2 \over 4}=A \\=>{dA \over da}=12-8a+{\sqrt3a \over 2}=0 \\=>a({\sqrt3 \over 2}-8)=-12 \\=>a={12 \over (8-{\sqrt3 \over 2})}$

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

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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.

taniska

2 Answers

Rinkoo Gupta

Arun Kumar

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