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label Calculus
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Package shipped by parcel post mail size restriction is that total of length and girth(distance around box perpendicular to the length)must be no more than 130 in.Suppose you are shipping rectangular box w/square end.What dimensions should max value?

Mar 19th, 2015

Probably you want to maximize volume of a parcel.

Denote length as L and height as h, then the depth is h also. Girth g = 4*h.

We know L + 4h <= 130, obviously we want L+4h = 130 (maximum allowed).

The volume V = L*h^2 = (130 - 4h)*h^2 = 130h^2 - 4h^3. Of course 0 <= h <= 130/4.

The maximum of a continuously differentiable function is achieved at ends (0 and 130/4) or in a point where derivative equals 0.

dV(h) = 260h - 12h^2 = 0. h1 = 0, h2 = 260/12 = 130/6 which is < 130/4.

V(0) = V(130/4) = 0 (not interesting). And V(h2) = (130 - 4*(130/6))*(130/6)^2 = 130*(1/3)*130^2/36 = 130^3/108 = approx. 20342,6 (in^3). It is > 0 and therefore a maximum.

Answer: h = 130/6 = 65/3 = 21,66...in, L = 130 - 4h = 130*(1 - 4/6) = 130/3 = 43,44...in.

Mar 19th, 2015

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