Postal regulations specify that a parcel sent by priority...

Postal regulations specify that a parcel sent by priority mail may have a combined length and girth of no more than 108 in. Find the dimensions of a rectangular package that has a square cross section and the largest volume that may be sent via priority mail. What is the volume of such a package? Hint: The length plus the girth is $4 x+h$ (see the accompanying figure).

Postal regulations specify that a parcel sent by priority mail may have a combined length and girth of no more than 108 in. Find the dimensions of a rectangular package that has a square cross section and the largest volume that may be sent via priority mail. What is the volume of such a package?
Hint: The length plus the girth is $4 x+h$ (see the accompanying figure).

Key Concepts

Constrained Optimization

This concept involves finding the maximum or minimum of a function subject to specific limitations or constraints. In many problems, including those related to dimensions of objects, an equation represents a constraint that must be respected, and the goal is to optimize a certain quantity (like volume) within that boundary.

Volume Maximization

Volume maximization refers to the process of determining the dimensions of a geometric object that yield the maximum volume under given conditions. It requires expressing the volume as a function of one or more variables and then using optimization techniques to find the best possible value.

Differentiation for Optimization

Differentiation is used to identify the critical points of a function, where its derivative equals zero or does not exist. These critical points are potential candidates for local maximums or minimums. In the context of maximizing volume, taking the derivative of the volume function with respect to a variable helps determine the optimal dimensions.

Geometric Modeling

Geometric modeling involves constructing a mathematical representation of a physical object. When dealing with packages or containers, it is common to define dimensions and relationships (such as a square cross section or the relation between length and girth) that are necessary to apply optimization techniques effectively.

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