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Minimizing Power Line Costs for a Factory Across a River

This problem explores the cost of running power lines from a power station across a river to a factory downriver. The costs differ for land and underwater installation, leading to a piecewise-defined cost function. By expressing the total cost as a function of the distance (x) where the lines go underwater, we analyze the function to find its domain and the optimal distance to minimize costs. Through graphing and application of the MINIMUM command, we determine that approximately 3.55 km of land lines minimizes costs effectively.

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Minimizing Power Line Costs for a Factory Across a River

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  1. 4.4Models Involving the Square Root Function and Piecewise-defined Functions

  2. A power station is on one side of the river that is 1/2km wide. A factory is 4 kilometers downstream on the other side of the river. It costs $10/100meters to run the power lines on the land and $15/100meters to run them under water. (a) Express the cost C of running the power lines from the power station to the factory as a function of x, where x is the distance from the factory to the point where the lines go under water. (see figure). 0.5km 4-x power station x factory

  3. The power lines have to cover a certain distance on the land And a distance under water. So the cost function is given by:

  4. (b) Find the domain of the cost function. Since the C(x) involves a radical We need Both quantities of this expression are squares, so the expression is always positive. Domain is all real numbers. Since it is an application problem, the feasible domain is 0 <x< 4.

  5. (c) Find x that would minimize the cost. That is find how many kilometers of land pipes are needed to minimize the cost. Graph the function C(x) and use the MINIMUM command to find the point (3.55, 45.59).

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