A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off squares from each corner and folding up the flaps. What should be side of the square to be cut off so that the volume of he box is maximum?

Let x (0 < x < 12) cm, be the length of each side of the square which is to be cut from each corner of the rectangular tin sheet of size 45 cm by 24 cm. Let V the volume of the open box formed by folding up the flaps.

therefore space space space straight V space equals straight x left parenthesis 45 minus 2 straight x right parenthesis thin space left parenthesis 24 minus 2 straight x right parenthesis space equals space 2 left parenthesis 2 straight x cubed minus 69 straight x squared plus 540 straight x right parenthesis space space space space space dV over dx space equals space 2 left parenthesis 6 straight x squared minus 138 straight x plus 540 right parenthesis space space space space space dV over dx space equals space 0 space space space space space space space rightwards double arrow space space space 2 left parenthesis 6 straight x squared minus 138 straight x plus 540 right parenthesis space equals space 0 space space space space space space rightwards double arrow space space space straight x cubed minus 23 straight x plus 90 space equals space 0 space space rightwards double arrow space space space space space left parenthesis straight x minus 5 right parenthesis thin space left parenthesis straight x minus 18 right parenthesis space equals space 0 space space space space space space rightwards double arrow space space space space straight x space equals space 5 comma space space 18

space space rightwards double arrow space space space space space straight x space equals space 5

open square brackets because space space straight x space equals space 18 space not an element of space left parenthesis 0 comma space 12 right parenthesis close square brackets

fraction numerator straight d squared straight V over denominator dx squared end fraction space equals space 2 left parenthesis 12 straight x minus 138 right parenthesis

At space straight x space equals space 5 comma space space space fraction numerator straight d squared straight V over denominator dx squared end fraction space equals space 2 left parenthesis 60 minus 138 right parenthesis space equals space minus 156 less than space 0 space space space space space rightwards double arrow space space space straight V space has space local space maximum space at space straight x space equals space 5

But x = 5 is the only extreme point

therefore space space space space space straight V space is space maximum space when space straight x space equals space 5 rightwards double arrow space space space space side space of space the space square space equals space 5 space cm

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