Time and Distance To Interception

A boat starts from A  
\[5 \mathbf{i} + 12 \mathbf{j}\]
  with a speed of 12 km/h on a bearing of 70 degrees. Another boat starts from the origin with a speed of 15 km/h to intercept the first ship. How long will it be before interception takes place, and where will interception take place?
The diagram illustrates the problem.


The first ship travels a distance 12T and the second ship travels a distance 15T before interception. The angle at A is  
  to 1 decimal place.
The distance of A from the origin is  
The Cosine Rule gives



\[T=\frac{211.2 \pm \sqrt{(-211.2)^2-4(81)(-169)}}{2 \times 81} = -0.64, \; 3.25 \]
Obviously the solution is 3.25 hours hours, and interception takes place 15(3.25)=48.75 km from the origin.

Add comment

Security code