Optimizing Time

Problem: Maya is 2 km offshore on a boat and wishes to reach a coastal village which is 6 km down the straight shoreline from the point on the shore nearest to the boat. She can row at 2 km/hour and run 5 km/hour. Where should she land her boat to reach the village in the least amount of time? To get an idea of what this question is asking drag the point P along the shoreline to see how the total time taken to complete the trip changes.
Read the problem at the top of the page.
  1. Drag P to 0 (the longest distance). How long does it take to get to the house?
  2. Drag P to 6 (the shortest distance). How long does it take to get to the house?
  3. Explain why the above are the shortest and longest distances?
  4. Use the applet to find the shortest time to get to the house. Check with the "optimal traveler". Were you correct?
  5. Let the distance from (0, 0) to P be x, what is the distance to the house from P?
  6. Based on the given rate, how long does it take to travel this distance. Recall that
  7. Find an equation, in terms of x, for the time it takes to travel the dashed blue line.
  8. Find an equation for the total time it takes to travel to the house.
Thanks to J Mulholland for creating the applet and question.