JEE Advanced 2013 Paper 1

l = 8x , b = 15x
∴ volume = (8x - 2a) (15x - 2a) (a) = 4a3−46a2x+120ax2

dv

da

=6a2−46ax+60x2
(

dv

dx

)at‌x=5=0
∴ x = 3 and

5

6

d2v

da2

=6a−23x
(

d2v

da2

)at‌a=5‌&‌x=3<0
So, at x = 3 gives maxima
(

d2v

da2

)at‌a=5‌&‌x=

5

6

>0
So, at x =

5

6

gives maxima

dv

da

=0 when a = 5 given (∴ 4a2 = 100 given for maximum volume) at a = 5
by

dv

da

= 0
⇒ 6x2 - 23x + 15 = 0
x = 3 or

5

6

So by x = 3 (for max volume)
8x = 24, 15x = 45