Q. 31

# A square piece of

Given,

Side of the square piece is 18 cms.

the volume of the formed box is maximum. Let us consider,

‘x’ be the length and breadth of the piece cut from each vertex of the piece.

Side of the box now will be (18-2x)

The height of the new formed box will also be ‘x’.

Let the volume of the newly formed box is :

V = (18-2x)2 × (x)

V = (324+ 4x2 – 72x) x

V = 4x3 -72x2 +324x ------ (1)

For finding the maximum/ minimum of given function, we can find it by differentiating it with x and then equating it to zero. This is because if the function V(x) has a maximum/minimum at a point c then V’(c) = 0.

Differentiating the equation (1) with respect to x:  -------- (2)

[Since ]

To find the critical point, we need to equate equation (2) to zero. x2 – 12x + 27 = 0  x = 9 or x =3

x= 2

[as x = 9 is not a possibility, because 18-2x = 18-18= 0]

Now to check if this critical point will determine the maximum area of the box, we need to check with second differential which needs to be negative.

Consider differentiating the equation (3) with x:  ----- (4)

[Since ]

Now let us find the value of As , so the function V is maximum at x = 3cm

Now substituting x = 3 in 18 – 2x, the side of the considered box:

Side = 18-2x = 18 - 2(3) = 18-6= 12cm

Therefore side of wanted box is 12cms and height of the box is 3cms.

Now, the volume of the box is

V = (12)2 × 3 = 144 × 3 = 432cm3

Hence maximum volume of the box formed by cutting the given 18cms sheet is 432cm3 with 12cms side and 3cms height.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation view all courses RELATED QUESTIONS :

If the sum of theMathematics - Board Papers

A metal box with Mathematics - Board Papers

Show that aRD Sharma - Volume 1

Find the local maMathematics - Board Papers

Prove that the seMathematics - Board Papers

Prove that the raMathematics - Board Papers

Prove that the leMathematics - Board Papers