1.  A rectangle is inscribed in a semicircle of radius r with one of its side on the diametetof the semicircle .  Find the dimensions of rectangle so that its ares is max. Find also its area.

Asked by poojaparmar065 | 11th Oct, 2014, 06:24: AM

Expert Answer:

L e t space x space b e space t h e space l e n g t h space o f space t h e space r e c tan g l e space a n d space l e t space y space b e space t h e space b r e a d t h space o f space t h e space r e c tan g l e. G i v e n space t h a t space apostrophe r apostrophe space i s space t h e space r a d i u s space o f space t h e space s e m i c i r c l e.
F r o m space t h e space f i g u r e comma space i t space i s space c l e a r space t h a t r squared equals open parentheses x over 2 close parentheses squared plus y squared space space space space space space space left square bracket B y space P y t h a g o r a s space T h e o r e m right square bracket rightwards double arrow r squared equals x squared over 4 plus y squared rightwards double arrow y squared equals r squared minus x squared over 4 rightwards double arrow y equals square root of r squared minus x squared over 4 end root A r e a space o f space t h e space r e c tan g l e equals x y rightwards double arrow A equals x square root of fraction numerator 4 r squared minus x squared over denominator 4 end fraction end root D i f f e r e n t i a t i n g space A space w i t h space r e s p e c t space t o space x comma space w e space h a v e comma fraction numerator d A over denominator d x end fraction equals fraction numerator x open parentheses minus 2 x close parentheses over denominator 2 square root of 4 r squared minus x squared end root end fraction plus fraction numerator square root of 4 r squared minus x squared end root over denominator 2 end fraction E q u a t i n g space t h e space a b o v e space f i r s t space d e r i v a t i v e space t o space z e r o comma space w e space h a v e rightwards double arrow fraction numerator x open parentheses minus 2 x close parentheses over denominator 2 square root of 4 r squared minus x squared end root end fraction plus fraction numerator square root of 4 r squared minus x squared end root over denominator 2 end fraction equals 0 rightwards double arrow minus 2 x squared plus 4 r squared minus x squared equals 0 rightwards double arrow 4 r squared equals 3 x squared rightwards double arrow x squared equals fraction numerator 4 r squared over denominator 3 end fraction rightwards double arrow x equals square root of fraction numerator 4 r squared over denominator 3 end fraction end root rightwards double arrow x equals fraction numerator 2 r over denominator square root of 3 end fraction ; y equals square root of r squared minus fraction numerator fraction numerator 4 r squared over denominator 3 end fraction over denominator 4 end fraction end root equals square root of fraction numerator 2 r squared over denominator 3 end fraction end root equals square root of 2 over 3 end root r T h u s comma space A equals x y equals fraction numerator 2 r over denominator square root of 3 end fraction cross times square root of 2 over 3 end root r rightwards double arrow A equals fraction numerator 2 square root of 2 over denominator 3 end fraction r squared space s q u a r e space u n i t s.

Answered by Vimala Ramamurthy | 11th Oct, 2014, 03:36: PM