Q. 184.0( 45 Votes )
A rectangular she
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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