Anett’s Island Problem

1. Anett’s Island Problem Kathryn Amejka

2. Dave lives at a house in California and Anett lives on her own island off the coast. Dave wants to build a path of lights poles all the way to Anett’s island so they can always find their way to each other.

3. Her island is 1,700 meters off the coastline and needs to be connected to Dave’s house 3,000 meters down the coast by light poles.

4. It costs 50 dollars per meter to place the light poles on the coastline and 80 dollars per meter to place the light poles in the water. How should the light poles be laid out to minimize the cost?

5. Picture Representation

6. Solving the Problem

7. Solving the Problem

8. Solving the Problem

9. Solving the Problem

10. Checking the Minimum: Graphically 1361.0897 appears as a minimum when found using a calculator.

11. Checking the Minimum: Using Calculus

12. Justifying the Minimum The value determined to be x is indeed a minimum because when the second derivative test is used, the result has a sign that is positive. This means that the graph is concave up, so the point is a minimum.

13. Conclusion Dave should have the light path laid out 1,638.91 meters from his house and then straight across to Anett’s island.