Show that all the rectangle inscribed for a given fixed circle the square has the maximum value?

Asked by Sonu | 3rd Sep, 2015, 10:36: PM

Expert Answer:

C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.
C o n s i d e r space a space c i r c l e space o f space r a d i u s space apostrophe r apostrophe. rightwards double arrow B D equals d i a m e t e r equals 2 r L e t space A B C D space b e space a space r e c tan g l e space i n s c r i b e d space i n space t h e space c i r c l e. L e t space t h e space w i d t h space o f space t h e space r e tan g l e space b e space apostrophe x apostrophe space a n d space l e n g t h space o f space t h e space r e c tan g l e space b e space apostrophe y apostrophe. C l e a r l y space B D space i s space t h e space d i a g o n a l space a n d space t h e space t r i a n g l e space A B D space i s space t h e space r i g h t space t r i a n g l e. B y space a p p l y i n g space P y t h a g o r a s comma space w e space h a v e comma space x squared plus y squared equals open parentheses 2 r close parentheses squared rightwards double arrow x squared plus y squared equals 4 r squared rightwards double arrow y squared equals 4 r squared minus x squared rightwards double arrow y equals square root of 4 r squared minus x squared end root W e space k n o w space t h a t space t h e space a r e a space o f space t h e space r e c tan g l e equals L e n g t h cross times B r e a d t h. rightwards double arrow A equals square root of 4 r squared minus x squared end root cross times x D i f f e r e n t i a t i n g space t h e space a b o v e space f u n c t i o n 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 cross times open parentheses negative 2 x close parentheses over denominator 2 square root of 4 r squared minus x squared end root end fraction plus square root of 4 r squared minus x squared end root cross times 1 rightwards double arrow fraction numerator d A over denominator d x end fraction equals fraction numerator negative 2 x squared over denominator 2 square root of 4 r squared minus x squared end root end fraction plus square root of 4 r squared minus x squared end root rightwards double arrow fraction numerator d A over denominator d x end fraction equals fraction numerator negative x squared over denominator square root of 4 r squared minus x squared end root end fraction plus square root of 4 r squared minus x squared end root rightwards double arrow fraction numerator d A over denominator d x end fraction equals fraction numerator negative x squared plus 4 r squared minus x squared over denominator square root of 4 r squared minus x squared end root end fraction rightwards double arrow fraction numerator d A over denominator d x end fraction equals fraction numerator 4 r squared minus 2 x squared over denominator square root of 4 r squared minus x squared end root end fraction... left parenthesis 1 right parenthesis F o r space m a x i m u m space o r space m i n i m u m comma space w e space h a v e comma space fraction numerator d A over denominator d x end fraction equals 0 rightwards double arrow fraction numerator 4 r squared minus 2 x squared over denominator square root of 4 r squared minus x squared end root end fraction equals 0 rightwards double arrow 4 r squared minus 2 x squared equals 0 rightwards double arrow 4 r squared equals 2 x squared rightwards double arrow x squared equals 2 r squared rightwards double arrow x equals square root of 2 r N o w space d i f f e r e n t i a t i n g space t h e space e q u a t i o n 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 squared A over denominator d x squared end fraction equals fraction numerator square root of 4 r squared minus x squared end root cross times open parentheses negative 4 x close parentheses minus open parentheses 4 r squared minus 2 x squared close parentheses cross times fraction numerator open parentheses negative 2 x close parentheses over denominator 2 square root of 4 r squared minus x squared end root end fraction over denominator open parentheses square root of 4 r squared minus x squared end root close parentheses squared end fraction rightwards double arrow fraction numerator d squared A over denominator d x squared end fraction equals fraction numerator open parentheses 4 r squared minus x squared close parentheses cross times open parentheses negative 4 x close parentheses minus open parentheses 4 r squared minus 2 x squared close parentheses cross times open parentheses negative x close parentheses over denominator open parentheses 4 r squared minus x squared close parentheses square root of 4 r squared minus x squared end root end fraction rightwards double arrow fraction numerator d squared A over denominator d x squared end fraction equals fraction numerator negative 4 x open parentheses 4 r squared minus x squared close parentheses plus x open parentheses 4 r squared minus 2 x squared close parentheses over denominator open parentheses 4 r squared minus x squared close parentheses square root of 4 r squared minus x squared end root end fraction rightwards double arrow fraction numerator d squared A over denominator d x squared end fraction equals fraction numerator negative 3 x open parentheses 4 r squared minus x squared close parentheses over denominator open parentheses 4 r squared minus x squared close parentheses square root of 4 r squared minus x squared end root end fraction rightwards double arrow fraction numerator d squared A over denominator d x squared end fraction equals fraction numerator negative 3 x over denominator square root of 4 r squared minus x squared end root end fraction less than 0 T h u s comma space a t space x equals square root of 2 r comma space t h e space a r e a space i s space m a x i m u m. S u b s t i t u t i n g space t h e space v a l u e space x equals square root of 2 r space i n space y equals square root of 4 r squared minus x squared end root comma space w e space h a v e comma y equals square root of 4 r squared minus 2 r squared end root equals square root of 2 r squared end root equals square root of 2 r space T h u s comma space x equals y space a n d space h e n c e space i t space i s space a space s q u a r e. T h u s comma space t h e space s q u a r e space h a s space t h e space m a x i m u m space v a l u e.

Answered by Vimala Ramamurthy | 4th Sep, 2015, 09:35: AM