IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

Parthav_knows_best

A large window has the shape of a rectangle surmounted by an equilateral triangle .

If the perimeter of the windows is 12 metres find the dimensions of the rectangle that

will produce largest area of windows .

answer is 12 / (6-square root of 3 ), (24 - 6square root of 3 ) / ( 6-square root of 3)

shine

see
let d side of equilateral triangle b 'x'
thus 1 of d side of rectangle common to triangle nd rect is x
let d other two be y
acccording to ques
total perimter = 3x+2y=12
=> y= (12-3x)/2

now total area of window= area of 3angle+ ar rectangle
= root 3/4 x^2 + xy
=root 3/4 x^2 + x(12-3x)/2

for max area
d(area)/dx = 0
=>root3 x - 6x +12=0
=>x= 12/ (6-root 3)
bt y= 18-6 root 3/(6- root 3)

.......wher's d mistake?????

varshavallig

shine ,ur rite.i worked it and got the same ans.may b the ans given is wrong.Parthav plz check it again.

!!!!!!!!!!!!!!!!!!!!!!!!!

spideyunlimited

Exactly!
Let the width of the window = a
and length of window = b
Thus
3a + 2b = 12

and Area = ab + sqrt(3).a^2/4
differentiate and equate to 0 to find critical point
we get a = 12 / [6 - sqrt(3)]
so area is maximum at this value of a.

therefore b can be found from
3a + 2b = 12
b = 18 - 6.sqrt(3) / [ 6 - sqrt(3)]

*in the solution given by you parthav, the length of the window is incorrect*

Parthav_knows_best

thank you

ya I too think the answer is given wrong here in the book .

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