A pipeline is to be constructed from a refinery across a...

A pipeline is to be constructed from a refinery across a swamp to storage tanks. See FIGURE 4.8.29. The cost of construction is $\$ 25,000$ per mile over the swamp and $\$ 20,000$ per mile over land. How should the pipeline be made so that the cost of construction is a minimum?

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Key Concepts

Optimization

Optimization involves finding the best solution from a set of alternatives, generally by maximizing or minimizing an objective function. In the context of this problem, optimization is used to determine the configuration that minimizes the total pipeline construction cost.

Cost Function Formulation

Cost function formulation is the process of expressing the total cost in terms of one or more variables that describe different segments or scenarios. Here, it involves setting up a function that adds the costs of building the pipeline over the swamp and over land, with each segment having its own cost per mile.

Calculus Differentiation

Calculus differentiation is used to analyze how the cost function changes with respect to the chosen variable. By taking the derivative of the cost function and setting it equal to zero, one can find the critical points that potentially minimize the cost.

Geometric Relationship

The geometric relationship pertains to using geometric principles, such as the Pythagorean theorem, to relate various distances in the problem. This allows for expressing one part of the pipeline's distance in terms of another, thereby integrating all segments into a single cost function.

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