Question

A lighthouse 30 meters high stands at

the top of a high cliff. The line of
sight from a point X on the bow of a
ship to point A at the top of the light-
house is at an angle of 18° to the hor-
izontal. The line of sight from point X
to point B at the bottom of the light-
house makes an angle of 14° with a
horizontal line. Draw a sketch of the
situation and then determine the
approximate height of the cliff.

Answer 1

Let's call the height of the cliff "h". From the sketch, we can see that the distance from point X to point B is the same as the height of the lighthouse, which is 30 meters.

We can use trigonometry to set up two equations involving h:

tan(18°) = h/d, where d is the horizontal distance from X to A
tan(14°) = (h + 30)/d

We can solve for d in the first equation by multiplying both sides by d and dividing by tan(18°):

d = h/tan(18°)

We can substitute this expression for d into the second equation and solve for h:

tan(14°) = (h + 30)/(h/tan(18°))
tan(14°) = (h + 30)tan(18°)/h
htan(14°) = (h + 30)tan(18°)
htan(14°) = htan(18°) + 30tan(18°)
h*(tan(14°) - tan(18°)) = 30tan(18°)
h = 30tan(18°)/(tan(14°) - tan(18°))

Using a calculator, we can evaluate this expression to get:

h ≈ 76.7 meters

Therefore, the approximate height of the cliff is 76.7 meters.