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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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