Question

A small house was built on an island off a perfectly straight shoreline. The point B on the shoreline that is closest to the island is exactly 6 miles from the island. Eight miles east of B is the closest source of fresh water to the island. A pipeline is to be built from the island to the source of fresh water by laying pipe underwater in a straight line from the island to a point Q on the shoreline between B and the water source, and then laying pipe on land along the shoreline from Q to the source. It costs 6000 dollars per mile to lay the pipe under the water and 3750 dollars per mile to lay pipe on land. How far east of B should Q be located in order to minimize the total construction costs

Answer 1

The question is a high school mathematics problem involving the optimization of pipeline construction costs in a geometry and calculus context, where the optimal point Q on the shoreline to minimize the total cost must be determined.

Step-by-step explanation:

The problem at hand is a mathematical optimization one, where the goal is to minimize the total construction costs of laying a pipeline from a small house on an island to a source of fresh water located 8 miles east of point B on the nearest shoreline. Determining the optimal location for point Q on the shoreline, which minimizes the total cost of laying the pipeline under the water and then along the shoreline, involves the application of the principles of geometry and calculus.

Specifically, the shortest path from the island to the shoreline is a perpendicular line to point B, which is 6 miles. To find the optimal point Q to which the pipeline should be laid under the water before going overland to the source, one needs to consider the difference in costs between underwater and onshore piping. Using calculus to minimize the cost function representing the total construction costs will provide the exact location of point Q east of B.