Answer :

Given,

• Side of the square piece is 12 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 (12-2x)

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

Let the volume of the newly formed box is :

V = (12-2x)^{2} × (x)

V = (144 + 4x^{2} – 48x) x

V = 4x^{3} -48x^{2} +144x ------ (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 f(x) has a maximum/minimum at a point c then f’(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.

x^{2} – 8x +12 = 0

x = 6 or x =2

x= 2

[as x = 6 is not a possibility, because 12-2x = 12-12= 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 A is maximum at x = 2

Now substituting x = 2 in 12 – 2x, the side of the considered box:

Side = 12-2x = 12 - 2(2) = 12-4= 8cms

Therefore side of wanted box is 8cms and height of the box is 2cms.

Now, the volume of the box is

V = (8)^{2} × 2 = 64 × 2 = 128cm^{3}

Hence maximum volume of the box formed by cutting the given 12cms sheet is 128cm^{3} with 8cms side and 2cms height.

Rate this question :

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