Shape and Space 4

4 marks · Hard

A ship is sailing directly towards a lighthouse. From one point at sea, the angle of elevation from the ship to the top of the lighthouse is 14°. After the ship sails 200 metres closer to the lighthouse in a straight line, the angle of elevation increases to 22°. The height of the lighthouse is h metres.

Calculate the height h of the lighthouse.

Give your answer correct to 3 significant figures.

Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.

Show answer

h = 92.7 m (to 3 s.f.) m (±0.1)

Common mistakes: Using sine or cosine ratio instead of tangent; Incorrect labelling of distances; Rounding error

Show mark scheme and worked solution

Marking points

M1 · 1 mark · Let distance from starting point to base of lighthouse be x. Label two right triangles, set up equations using tangent ratio for both positions.

M2 · 1 mark · Form two equations: tan(14°) = h/(x+200); tan(22°) = h/x. Set up simultaneous equations to eliminate h and solve for x.

A1 · 1 mark · Correctly solve for x and substitute back to find h. (follow-through)

A2 · 1 mark · Give final answer for h correct to 3 significant figures, with unit. (follow-through)

Worked solution

Let the starting point be S, and the distance from S to the base of the lighthouse be x metres.

At the first point, tan(14°) = h / (x + 200)

At the second point, tan(22°) = h / x

Equate the two expressions for h:

Expand and solve for x:

Calculate x:

Now substitute x into h = x tan 22°:

But check method; earlier solution logic shows (the correct process has the ship getting closer, so initial position S is farther away, so x is the distance from the second position to the lighthouse, i.e. the nearest point, and x+200 is the starting point).

However, correcting the associated geometry: In reality, the ship is 200 m closer after the second measurement, which affects which distance is associated with which angle.

Let x be the distance from the second position (closer). Then tan(14°) = h / (x + 200), tan(22°) = h / x.

So h = x tan 22°, h = (x+200) tan 14° -> x tan 22° = (x+200) tan 14° -> x (tan 22° - tan 14°) = 200 tan 14° -> x = (200 tan 14°) / (tan 22° - tan 14°).

Final answer: h = 130 m (to 3 s.f.).