Case Study 2:

A lighthouse stands tall on a cliff by the sea, watching over ships that pass by. One day a ship is seen approaching the shore and from the top of the lighthouse, the angles of depressi

Question

Case Study 2:

A lighthouse stands tall on a cliff by the sea, watching over ships that pass by. One day a ship is seen approaching the shore and from the top of the lighthouse, the angles of depression of the ship are observed to be $30°$ and $45°$ as it moves from point P to point Q. The height of the lighthouse is 50 metres.

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Based on the information given above, answer the following questions:
(i) Find the distance of the ship from the base of the lighthouse when it is at point Q, where the angle of depression is $45°$.
(ii) Find the measures of $\angle PBA$ and $\angle QBA$.
(iii) (a) Find the distance travelled by the ship between points P and Q.
**OR**
(b) If the ship continues moving towards the shore and takes 10 minutes to travel from Q to A, calculate the speed of the ship in km/h, from Q to A.

Accepted Answer

Step 1: (i) $\angle AQB = \angle QBX = 45°$ and $\angle APB = \angle PBX = 30°$
In $	riangle AQB$, $	an 45° = \frac{50}{AQ}$
$AQ = 50$ m
Step 2: (ii) $\angle PBA = 60°$
$\angle QBA = 45°$
Step 3: (iii)(a) In $	riangle APB$, $	an 30° = \frac{50}{AP}$
$AP = 50\sqrt{3}$ m
Distance travelled by the ship $= PQ = 50\sqrt{3} - 50 = 50(\sqrt{3} - 1)$ m or $36.5$ m
Step 4: **OR**
(iii)(b) Speed of the ship $= \frac{50 	ext{ metres}}{10 	ext{ minutes}} = 0.3$ km/h

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