Answer:
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:
[tex]time=\frac{distance}{speed}[/tex]
Then the total travel time from the yacht to the hospital will be the motorboat travel time plus the ambulance travel time
[tex]t=t_m+t_a[/tex]
[tex]t=\frac{d_m}{s_m} +\frac{d_a}{s_a}[/tex]
First we must write the total travel time in terms of the motorboat's direction (Θ).
[tex]cos(\theta)=\frac{15}{d_m}[/tex]
[tex]d_m=\frac{15}{cos(\theta)}=15 sec(\theta)[/tex]
[tex]d_a=60-d_1[/tex]
[tex]tan(\theta)=\frac{d_1}{15}[/tex]
[tex]d_1=15tan(\theta)[/tex]
[tex]d_a=60-15tan(\theta)[/tex]
[tex]t=t_m+t_a[/tex]
[tex]t=\frac{d_m}{s_m} +\frac{d_a}{s_a}[/tex]
[tex]t=\frac{15sec(\theta)}{20} + \frac{60-15tan(\theta)}{90}[/tex]
[tex]t=\frac{15}{20}sec(\theta) + \frac{60}{90}-\frac{15}{90}tan(\theta)[/tex]
[tex]t=\frac{3}{4}sec(\theta)-\frac{1}{6}tan(\theta) + \frac{2}{3}[/tex]
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 [tex]\frac{dt}{d\theta}[/tex] and then find its roots (its x-interceptions).
[tex]\frac{dt}{d\theta}=\frac{d}{d\theta} (\frac{3}{4}sec(\theta)-\frac{1}{6}tan(\theta) + \frac{2}{3})[/tex]
[tex]\frac{dt}{d\theta}=\frac{3}{4}sec(\theta)tan(\theta)-\frac{1}{6}sec^2(\theta)[/tex]
[tex]\frac{dt}{d\theta}=sec(\theta)(\frac{3}{4}tan(\theta)-\frac{1}{6}sec(\theta))[/tex]
Now let's find the values of x which make [tex]\frac{dt}{d\theta}=0[/tex]
[tex]\frac{dt}{d\theta}=sec(\theta)(\frac{3}{4}tan(\theta)-\frac{1}{6}sec(\theta))=0[/tex]
As sec(\theta) is never equal to zero, then [tex]\frac{dt}{d\theta}[/tex] would be zero when
[tex]\frac{3}{4}tan(\theta)=\frac{1}{6}sec(\theta)[/tex]
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 [tex]t(\theta)[/tex]. 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.
