Walking and rowing A boat on the ocean is 4 mi from the nearest point on a straight shoreline; that point is 6 mi from a restaurant on the shore. A woman plans to row the boat straight to a point on the shore and then walk along the shore to the restaurant. a. If she walks at 3 mi/hr and rows at 2 mi/hr, at which point on the shore should she land to minimize the total travel time? b. If she walks at 3 mi/hr, what is the minimum speed at which she must row so that the quickest way to the restaurant is to row directly (with no walking)?
Solution Step 1: A boat is 4 miles from the nearest point to the shore, that point is 6 miles from the restaurant. A person plans to row the boat straight and walk along the shore to the restaurant. The following diagram describes this situation. Step 2: (a)If a person rows at and walks then determine the point on the shore that the person land to minimize the total travel time. Step 3 The distance between boat and shoreline is, The distance between shoreline and restaurant is, The speed of the boat is, The walking speed is, Assume that the rowing distance is x and the walking distance from restaurant to opposite to shore is y . So the walking distance is Step 4 The objective function is the total travel time s which is the sum of the time taken for rowing the boat and walking along the shoreline. The travel time is the distance travelled divide by the speed. The length of the rowing path is, So, the time taken for the rowing is, Step 5 The length of walking path is, Therefore, the time taken for the walking is, Step 6 The total time for the trip is, To minimize the objective function, find the critical points of T. Hence, to minimize the total travel time the point on the shore at