CBSE Class 12-science Answered

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

Asked by Adeeb Khan | 21 Jul, 2014, 02:03: PM

Expert Answer

Consider the following figure.

$S i n c e space t h e space t r i a n g l e space capital delta A B C space i s space a n space i s o s c e l e s space t r i a n g l e comma space t h e space p e r p e n d i c u l a r space f r o m space t h e space v e r t e x space A space t o space t h e space b a s e space B C space b i s e c t s B C. T h e r e f o r e comma space B D equals D C equals 9 minus x squared B y space P y t h a g o r a s space T h e o r e m comma A C squared equals A D squared plus C D squared rightwards double arrow open parentheses 9 plus x squared close parentheses squared equals A D squared plus open parentheses 9 minus x squared close parentheses squared rightwards double arrow 81 plus x to the power of 4 plus 18 x squared equals A D squared plus 81 plus x to the power of 4 minus 18 x squared rightwards double arrow A D squared equals 36 x squared rightwards double arrow A D equals 6 x rightwards double arrow h e i g h t equals 6 x T h e space a r e a space o f space t h e space t r a i n g l e space A B C comma space A equals 1 half cross times B C cross times A D rightwards double arrow A equals 1 half cross times open parentheses 18 minus 2 x squared close parentheses cross times 6 x rightwards double arrow A equals open parentheses 18 minus 2 x squared close parentheses cross times 3 x rightwards double arrow A equals 54 x minus 6 x cubed.... left parenthesis 1 right parenthesis 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 54 minus 18 x squared...... left parenthesis 2 right parenthesis T o space f i n d space t h e space m a x i m a space a n d space m i n i m a comma space l e t space u s space e q u a t e space fraction numerator d A over denominator d x end fraction equals 0 rightwards double arrow 54 minus 18 x squared equals 0 rightwards double arrow 18 x squared equals 54 rightwards double arrow x squared equals 3 rightwards double arrow x equals plus-or-minus square root of 3 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 left parenthesis 2 right parenthesis space a g a i 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 minus 36 x less than 0 T h u s comma space a t space x equals square root of 3 comma space A r e a space A space i s space m a x i m u m. C o n s i d e r space t h e space e q u a t i o n space A equals 54 x minus 6 x cubed N o w space s u b s t i t u t e space x equals square root of 3 comma space i n space t h e space a b o v e space e q u a t i o n comma space w e space h a v e comma A equals 54 cross times square root of 3 minus 6 open parentheses square root of 3 close parentheses cubed rightwards double arrow A equals square root of 3 open parentheses 54 minus 6 cross times 3 close parentheses rightwards double arrow A equals square root of 3 open parentheses 54 minus 18 close parentheses rightwards double arrow A equals 36 square root of 3 s q. u n i t s$