Question

A girl standing on a lighthouse built on a cliff near the seashore, observes two boats due East of the lighthouse. The angles of depression of the two boats are 300 and 600. The distance between the boats is 300m. 

a. Draw a rough figure based on the given details. 

b. Find the distance of the top of the lighthouse from the sea level. (Boats and foot of the lighthouse are in a straight line).

Answer 1

b. In the right angled ∆ABD

\(tan60^\circ=\frac{AD}{AB}\)

\(AB=\frac{AD}{tan60^\circ}\)

\(x=\frac{h}{\sqrt{3}}\)

In the right angled ∆ACD

\(tan30^\circ=\frac{AD}{AC}\)

\(AC=\frac{AD}{tan30^\circ}\)

\(x+300=\frac{h}{tan30^\circ}\)

\(x+300=h\sqrt{3}\)

\(\therefore \frac{h}{\sqrt{3}}+300=h\sqrt{3}\)

\(h+300\sqrt{3}=3h\)

\(2h=300\sqrt{3}\)

\(h=150\sqrt{3}=259.8m\)