A  rectangular window  is surmounted by an equilateral triangle . Given that the  perimeter is16 cm., find the width of the window  so that the maximum amount of light is admitted .

Asked by inderjeet.saluja | 26th Aug, 2014, 11:52: PM

Expert Answer:

G i v e n space t h a t space p e r i m e t e r space i s space 16 space u n i t s. P e r i m e t e r space o f space t h e space r e c tan g u l a r space w i n d o w space s u r m o u n t e d space b y space a n space e q u i l a t e r a l space t r i a n g l e space equals 2 l plus b plus b plus b equals 16 rightwards double arrow 3 b equals 16 minus 2 l rightwards double arrow b equals fraction numerator 16 minus 2 l over denominator 3 end fraction A r e a space i s space g i v e n space b y A equals l b plus fraction numerator square root of 3 over denominator 4 end fraction b squared rightwards double arrow A equals l cross times fraction numerator 16 minus 2 l over denominator 3 end fraction plus fraction numerator square root of 3 over denominator 4 end fraction open parentheses fraction numerator 16 minus 2 l over denominator 3 end fraction close parentheses squared fraction numerator d A over denominator d l end fraction equals fraction numerator 16 minus 4 l over denominator 3 end fraction plus fraction numerator square root of 3 over denominator 4 end fraction open parentheses fraction numerator 16 minus 2 l over denominator 3 end fraction close parentheses open parentheses minus 2 close parentheses F o r space m a x i m u m comma space fraction numerator d A over denominator d l end fraction equals 0 fraction numerator 16 minus 4 l over denominator 3 end fraction plus fraction numerator square root of 3 over denominator 4 end fraction cross times 2 cross times open parentheses fraction numerator 16 minus 2 l over denominator 3 end fraction close parentheses open parentheses minus 2 close parentheses equals 0  fraction numerator 16 minus 4 l over denominator 3 end fraction equals fraction numerator square root of 3 over denominator 4 end fraction open parentheses fraction numerator 16 minus 2 l over denominator 3 end fraction close parentheses open parentheses 4 close parentheses fraction numerator 16 minus 4 l over denominator 3 end fraction equals square root of 3 open parentheses fraction numerator 16 minus 2 l over denominator 3 end fraction close parentheses 16 minus 4 l equals square root of 3 open parentheses 16 minus 2 l close parentheses 16 minus 16 square root of 3 equals 4 l minus 2 square root of 3 l 16 open parentheses 1 minus square root of 3 close parentheses equals 2 l open parentheses 2 minus square root of 3 close parentheses 8 open parentheses 1 minus square root of 3 close parentheses equals l open parentheses 2 minus square root of 3 close parentheses l equals fraction numerator 8 open parentheses 1 minus square root of 3 close parentheses over denominator 2 minus square root of 3 end fraction W e space k n o w space t h a t space b equals fraction numerator 16 minus 2 l over denominator 3 end fraction S u b s t i t u t i n g space t h e space v a l u e space o f space l equals fraction numerator 8 open parentheses 1 minus square root of 3 close parentheses over denominator 2 minus square root of 3 end fraction comma space w e space c a n space f i n d space t h e space v a l u e space o f space b

Answered by Vimala Ramamurthy | 27th Aug, 2014, 11:10: AM

Queries asked on Sunday & after 7pm from Monday to Saturday will be answered after 12pm the next working day.