Postal regulations specify that a parcel sent by priority mail may have a combin

Question

Postal regulations specify that a parcel sent by priority mail may have a combined length and girth of no more than 126 in. Find the dimensions of a rectangular package that has a square cross section and largest volume that may be sent by priority mail. Hint: The length plus the girth is 4x +l.

length     in width     in height     in Postal regulations specify that a parcel sent by priority mail may have a combined length and girth of no more than 126 in. Find the dimensions of a rectangular package that has a square cross section and largest volume that may be sent by priority mail. Hint: The length plus the girth is 4x +l. What is the volume of such a package?

Explanation / Answer

Let x = the length of a side of the square base.

Perimeter: P = 4x + h = 126 => h = 126-4x

Volume: V = x * x * h = x * x * (126-4x)
V = 126x^2 - 4x^3

To determine the maximum volume, we need to take the derivative and solve for where it is equal to zero:

V' = 252x - 12x^2 = 0

4x(63 - 3x) = 0

x = 0 or x = (63/3) in = 21 in

Since x=0 cm does not make physical sense with the problem, we reject it.

Therefore the dimensions that maximize volume are 21 in x 21 in x 42 in.

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