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Q:An open box with a square base is to be made out of a given quantity of metal sheet of area C squared .
Show that the maximum volume of cube is C cubed divided by 6 square root of 3.
 

Asked by Ravi consul 15th March 2015, 8:10 PM
Answered by Expert
Answer:
L e t space x space b e space t h e space s i d e space o f space t h e space s q u a r e space b a s e space a n d space y space b e space t h e space h e i g h t space o f space t h e space b o x T h e n space s u r f a c e space a r e a space o f space t h e space o p e n space b o x equals x squared plus 4 x y G i v e n space t h a t space t h e space s u r f a c e space a r e a equals C squared T h u s comma space C squared equals x squared plus 4 x y rightwards double arrow y equals fraction numerator C squared minus x squared over denominator 4 x end fraction... left parenthesis 1 right parenthesis V o l u m e space o f space t h e space o p e n space b o x comma space V equals x squared cross times y rightwards double arrow V equals x squared cross times fraction numerator C squared minus x squared over denominator 4 x end fraction rightwards double arrow V equals x cross times fraction numerator C squared minus x squared over denominator 4 end fraction 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 rightwards double arrow V equals fraction numerator x C squared minus x cubed over denominator 4 end fraction rightwards double arrow fraction numerator d V over denominator d x end fraction equals fraction numerator C squared minus 3 x squared over denominator 4 end fraction... left parenthesis 2 right parenthesis F o r space a space m a x i m u m space o r space m i n i m u m comma space w e space s h o u l d space h a v e comma space fraction numerator d V over denominator d x end fraction space equals 0 rightwards double arrow fraction numerator C squared minus 3 x squared over denominator 4 end fraction equals 0 rightwards double arrow C squared minus 3 x squared equals 0 rightwards double arrow 3 x squared equals C squared rightwards double arrow x squared equals C squared over 3 rightwards double arrow x equals plus-or-minus fraction numerator C over denominator square root of 3 end fraction rightwards double arrow x equals fraction numerator C over denominator square root of 3 end fraction comma space a s space s i d e space c a n n o t space b e space n e g a t i v e. 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 o n c e space a g a i n comma space w e space h a v e comma fraction numerator d squared V over denominator d x squared end fraction equals fraction numerator negative 3 over denominator 4 end fraction cross times 2 x rightwards double arrow open square brackets fraction numerator d squared V over denominator d x squared end fraction close square brackets subscript x equals fraction numerator C over denominator square root of 3 end fraction end subscript equals fraction numerator negative 3 over denominator 2 end fraction open parentheses fraction numerator C over denominator square root of 3 end fraction close parentheses less than 0 H e n c e space t h e space v o l u m e space i s space m a x i m u m space a t space x equals fraction numerator C over denominator square root of 3 end fraction F r o m space e q u a t i o n space left parenthesis 1 right parenthesis comma space y equals fraction numerator C squared minus x squared over denominator 4 x end fraction y equals fraction numerator C squared minus open parentheses fraction numerator C over denominator square root of 3 end fraction close parentheses squared over denominator 4 open parentheses fraction numerator C over denominator square root of 3 end fraction close parentheses end fraction equals fraction numerator begin display style fraction numerator 3 C squared minus C squared over denominator 3 end fraction end style over denominator begin display style fraction numerator 4 C over denominator square root of 3 end fraction end style end fraction equals fraction numerator 2 C squared over denominator 12 C end fraction cross times square root of 3 equals fraction numerator C square root of 3 over denominator 6 end fraction V o l u m e space o f space t h e space b o x equals x squared cross times y equals open parentheses fraction numerator C over denominator square root of 3 end fraction close parentheses squared cross times fraction numerator C square root of 3 over denominator 6 end fraction equals C squared over 3 cross times fraction numerator C square root of 3 over denominator 6 end fraction equals fraction numerator C cubed over denominator 6 square root of 3 end fraction space c u b i c space u n i t s.
Answered by Expert 16th March 2015, 12:10 PM
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