Answer :

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+ 4x^{2} – 72x) x

V = 4x^{3} -72x^{2} +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.

x^{2} – 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 = 432cm^{3}

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

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