As observed from the top of a 75 m high lighthouse from the sea level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.
Let AB be the lighthouse of height 75 m. Let C and D be the positions of the ships.
30° and 45° are the angles of depression from the lighthouse.
Draw a figure based on given instructions:
CD = distance between two ships
Step 1: From right triangle ABC,
tan 45° = AB/BC
1= 75/BC
BC = 75 m
Step 2: Form right triangle ABD,
tan 30° = AB/BD
1/√3 = 75/BD
BD = 75√3
Step 3: To find measure of CD, use results obtained in step 1 and step 2.
CD = BD – BC = (75√3 – 75)
= 75(√3-1)
The distance between the two ships is 75(√3-1) m.