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A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 114 inches (see figure). Find the dimensions of the package of maximum volume that can be sent. (Assume the cross section is square.)
X= ??????
Y= ???????

Sagot :

The dimensions of the rectangular package with the condition of maximum volume is equal to  x = 19inches and y = 38inches.

As given in the question,

Let us consider length of the cross section which is square be x

Perimeter of the cross section is 4x

And y be the length of the rectangular package

Maximum combined ( length + perimeter of the cross section ) = 114

y + 4x = 114

⇒ y = 114 -4x

Volume of the package 'V' = x²y

⇒V = x²(114 -4x)

⇒V = 114x² -4x³

Differentiate both the side of the equation with respect to x,

dV/dx = 228x - 12x²

For maximum Volume

dV/dx =0

228x -12x² = 0

⇒12x( 19 -x ) =0

⇒12x = 0 or 19 - x =0

⇒x =0 or x =19

x =0 is not possible

⇒x =19inches

y = 114 - 4(19)

  = 38inches

Therefore, the dimensions of the rectangular package with the given condition of volume is equal to x = 19inches and y = 38inches.

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