Maharashtra State BoardHSC Arts 12th Board Exam
Advertisement Remove all ads

An Open Box is to Be Made Out of a Piece of a Square Card Board of Sides 18 cms. by Cutting off Equal Squares From The Comers and Tumi 11g up the Sides. Find the Maximum Volume of the Box. - Mathematics and Statistics

Sum

An open box is to be made out of a piece of a square card board of sides 18 cms. by cutting off equal squares from  the comers and turning up the sides. Find the maximum volume of the box.

Advertisement Remove all ads

Solution


Let each side of the square cut off from each corner be x cm.
Then the base of the box will be of side 18 - 2x cm and the height of the box will be x cm
Then volume of box V = ( 18 - 2x )( 18 - 2x )
V = ( 18 - 2x )2x
V = 4x3 + 324x - 72x2                             ..(i)
Differentiating w.r t to x , we get
`(dV)/dx = 12x^2 + 324 - 144x`

`(dV)/dx = 12(x^2 - 12x + 27)`               ...(ii)
For maximum volume `(dV)/dx = 0`
⇒ 12( x2 - 12x + 27 ) = 0
⇒ x2 - 9x -3x + 27 = 0
⇒ ( x - 9 )( x - 3 ) = 0
⇒ x = 9, 3
Again differentiating, we get
`(d^2V)/dx^2 = 2x - 12`                         ....(iii)
At x = 9.
`(d^2V)/dx^2 = +ve`
 V is minimum at x = 9 at x = 3.
`(d^2V)/dx^2 = - ve`
 V is maximum at x = 3.
 Maximum volume V =
= 12 x 12 x 3 = 432 cm3

  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

Advertisement Remove all ads

Video TutorialsVIEW ALL [2]

Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×