Shortcut:
The angles of depression of two ships from the top of a lighthouse are θ_{1} and θ_{2}. If the ships are 'm' metres apart, then the,
(i) Height of the lighthouse =
mtanθ_{1}tanθ_{2}
/
tanθ_{1} + tanθ_{2}
metres,
(ii) Distance of ship at P from the foot of the lighthouse =
mtanθ_{1}
/
tanθ_{1} + tanθ_{2}
metres
(iii) Distance of ship at Q from the foot of lighthouse =
mtanθ_{2}
/
tanθ_{1} + tanθ_{2}
metres
Here, m = 100, θ_{1} = 30°, θ_{2} = 45°
Using these values in the shortcut, we get:
Height of tower =
100 x tan30° x tan45°
/
tan30° + tan45°
=
100 x tan30° x tan45°
/
tan30° + tan45°
=
100 x 1/√3 x 1
/
1/√3 + 1
=
100 x 1 x 1
/
1 + √3
=
100
/
1 + 1.732
=
100
/
2.732
= 36.6
Hence, the height of the house is 36.6 metres.
