The length of the package is 32 inches and the width and height of the package are 16 inches each.
The volume of the package is 8192 cubic inches.
(a)
Assume that the length is y in and that the width and height be equal to x in as the base is a square.
Therefore,
The girth is 4*x = 4x inches
The maximum sum of length and girth is 72.
So, according to the equation:
y + 4x = 72
Subtracting 4x from each side of the equation,
y + 4x - 4x = 72 - 4x
y = 72 - 4x
The following formula can be used to determine the package's volume:
Volume = length × width × height
Volume = y × x × x
Volume = x²y cubic inches
Substituting the values in the above equation,
Volume = x² ( 96 - 4x )
Volume = 96x² - 4x³ cubic inches
The volume will be maximum when dV / dx = 0.
So,
dV/dx = d/dx ( 96x² - 4x³ ) = 0
2(96)x - 4(3)x² = 0
192x - 12x² = 0
x( 192 - 12x ) = 0
So, x = 0 which is not possible.
or
192 - 12x = 0
Adding 12x on each side of the equation.
192 = 12x
Dividing each side of the equation by 12.
x = 16 inch
Now substituting this value in y = 96 - 4x.
y = 96 - 4x
y = 96 - 4(16)
y = 96 - 64
y = 32 inch
(b)
The volume of the package = length × width × height
volume = 32 × 16 × 16
volume = 8192 cubic inches
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