Alden, a passenger on a yacht moored 15 miles due north of a straight, east-west
coastline, has become ill and has to be taken ashore in a small motorboat, which will meet
an ambulance at some point on the shore. the ambulance will then take alden to the
hospital, which is 60 miles east of the shore point closest to the yacht. the motorboat can
travel at 20 mph and the ambulance at 90 mph. in what direction should the motorboat
head, to minimize the travel time to the hospital? express your answer using an angle.

To minimize the travel time from the yacht to the hospital, the motorboat should head in a direction of 12.83 degrees west of south.

Explanation:

If we assume that both the motorboat and ambulance will be moving at a constant speed, we can calculate the time that each one will take to travel a given distance using the following equation:

Then the total travel time from the yacht to the hospital will be the motorboat travel time plus the ambulance travel time

First we must write the total travel time in terms of the motorboat's direction (Θ).

So this last equation represents the variation of the total travel time as a function of the motorboat's direction.

To find the equation's minimum point (which would be the direction with the minimum total travel time), we must find and then find its roots (its x-interceptions).

Now let's find the values of x which make

As sec(\theta) is never equal to zero, then would be zero when

Graphing both equations we can find their interceptions and this would the value we're looking for.

In the attached images we can see that \theta=0.224 rad=12.83° is the minimum point for . Then, to minimize the travel time from the yacht to the hospital, the motorboat should head in a direction of 12.83 degrees west of south.