1 ) Show that the total surface area of a closed cuboid with square base and given volume , is minimum, when its cube
 
2)The lenghts of the sides of an isosceles triangle are 9+x^2,9+x^2 and 18-2x^2 units, calculate the area of the triangle in terms of x and find the value of x which makes the area maximum.
 
3) The sum of the perimeter of a circle and a square is K, where K is some constant. Prove that the sum of their areas is least when the side of the square is double the radius of circle?

Asked by Adeeb Khan | 19th Jul, 2014, 06:10: PM

Expert Answer:

1. space L e t space l e n g t h space equals b r e a d t h equals l space & space h e i g h t space equals h V o l u m e comma V equals l cross times l cross times h rightwards double arrow h equals V over l squared T S A comma S equals 2 left parenthesis l squared plus l h plus l h right parenthesis space space space space space space space equals 2 left parenthesis l squared plus 2 l h right parenthesis space space space space space space space equals 2 open parentheses l plus 2 l V over l squared close parentheses space space space space space space space space S equals 2 open parentheses l plus fraction numerator 2 V over denominator l end fraction close parentheses D i f f r e n t i a t i n g space w r t space l comma fraction numerator d S over denominator d l end fraction equals 2 open parentheses 1 minus V over l squared close parentheses F o r space m a x i m a space o r space m i n i m a comma fraction numerator d S over denominator d l end fraction equals 0 rightwards double arrow 2 open parentheses 1 minus V over l squared close parentheses equals 0 rightwards double arrow open parentheses 1 minus V over l squared close parentheses equals 0 rightwards double arrow V over l squared equals 1 rightwards double arrow V equals l squared T h e r e f o r e comma space V equals l squared cross times l equals l cubed fraction numerator d squared S over denominator d l end fraction equals 2 open parentheses fraction numerator 2 V over denominator l cubed end fraction close parentheses greater than 0 I t space i s space a space m i n i m u m space v a l u e. T h e r f o r e comma space t h e space c u b o i d space w i l l space h a v e space a space m i n i m u m t o t a l space s u r f a c e space a r e a space w h e n space i t space i s space a space c u b e.

Answered by Prasenjit Paul | 21st Jul, 2014, 11:22: AM