Answer :

Let the side of the square to be cut off be x, then, the height of the box is x and the length is 45-2x and the breadth is 24 – 2x.

Then, the volume {V(x)} of the box is given by:

V(x) = x(45-2x)(24-x)

= x(1080-90x-48x+4x2)

= 4x3 – 138x2 + 1080x

V’(x) = 12x2 – 276x + 1080

= 12(x2 - 23x + 90)

=12(x – 18)(x – 5)

Now, V’’(x) = 24x – 276 = 12(2x-23)

Now, V’(x) = 0

x = 18 or 5

It is not possible to cut off a square of side 18cm from each corner of the rectangular sheet. So, x cannot be equal to1 8.

Therefore, x =5

V’’(5) = 12(10 – 23) = -156 < 0

Then, by second derivative test, x = 5 is the point of maxima of V.

Therefore, the side of the square to be cut off to make the volume of the box maximum possible is 5cm.

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