# Thread: Optimization Question - General Coordinates?

1. ## Optimization Question - General Coordinates?

The question is this:

A yacht is travelling due WEST at 6km/h. 3km NORTH WEST of the boats location, there is a boat travelling 4km/h, SOUTH WEST. How close to eachother do the boats get.

From how I understand these questions, I needed to sub something into a distance forumla (distance between 2 coordinates)... I could sub in some form of general coordinates for each boat's path.... or I could sub in their position equations....

I don't know how to begin this question!!! I'm drowning in this stuff..

2. Originally Posted by mike_302
The question is this:

A yacht is travelling due WEST at 6km/h. 3km NORTH WEST of the boats location, there is a boat travelling 4km/h, SOUTH WEST. How close to eachother do the boats get.

From how I understand these questions, I needed to sub something into a distance forumla (distance between 2 coordinates)... I could sub in some form of general coordinates for each boat's path.... or I could sub in their position equations....

I don't know how to begin this question!!! I'm drowning in this stuff..
Use a coordinate system. Place the yacht at the time t = 0 at the origin. Draw a sketch of this situation! Then the way of the yacht is described by:

$Y:\left\{\begin{array}{l}x = -6t\\y=0\end{array}\right.$

The boat at t = 0 is at $B\left(-\frac32 \sqrt{2}\ ,\ \frac32 \sqrt{2}\right)$ (Use Pythagorean theorem on an isosceles triangle). The way of the boat is described by:

$B:\left\{\begin{array}{l}x= -\frac32 \sqrt{2} - t \\y = \frac32 \sqrt{2} - t\end{array}\right.$

Now calculate the distance betwee the yacht and the boat:

$d(Y,B)=\sqrt{(-\frac32 \sqrt{2}-t-(-6t))^2+(\frac32 \sqrt{2}-t-0)^2} = \sqrt{26t^2-18\sqrt{2} \cdot t +9}$

If $d(Y,B)$ has a minimum then $(d(Y,B))^2$ has a minimum too. Calculate the first derivation of $(d(Y,B))^2$ and solve the equation

$\left((d(Y,B))^2\right)' = 0$

for t. I've got $t \approx 0.4895\ h$

Plug in this value into the equation of the distance to answer your question.