Hello, Chez_!

A ship travelling with a constant speed and direction is signhted from a lighthouse.

At this time it's 2.7km away on a bearing of 42°.

Half an hour later, it is on a bearing of 115° at a distence of 7.6 km.

Find its speed in km per hour Code:

N A
: *
: 42° * \
: *2.7 \
: * \
L * 73° \ x
* \
* \
7.6* \
* \
* \
* B

Bearings are measured clockwise from North.

The lighthouse is at $\displaystyle L.$

The ship is first sighted when $\displaystyle \angle NLA = 42^o,\;LA = 2.7\text{ km}$

The ship is sighted next when $\displaystyle \angle NLB = 115^o\quad\Rightarrow\quad \angle ALB = 73^o$

. . and $\displaystyle LB = 7.6\text{ km}$

We want the distance: $\displaystyle x \:=\:AB$

In $\displaystyle \Delta ALB$, use the Law of Cosines:

. . $\displaystyle x^2 \;=\;2.7^2 + 7.6^2 - 2(2.7)(7.6)\cos73^o \;=\;53.05106524$

Hence: .$\displaystyle x \;=\;7.28361622\text{ km}$

Now you can calculate its speed, right?