A person sitting on a boat is situated 2m above sea level. They note that a straight line from themselves to the top most point of a nearby lighthouse is 16 degrees. After travelling a further 50m directly away from the lighthouse this angle has decreased to 12 degrees.

How high is the topmost point of the lighthouse above sea level?

Just figure the height above the people, and then add 2 meters.

Draw a diagram, and it will be clear that if h is the height above the boaters,

h cot12° - h cot16° = 50

To solve this problem, we can apply trigonometry. Let's break down the given information and work through the solution step by step.

1. A person on a boat is situated 2m above sea level.
2. The person notes that the angle from themselves to the topmost point of the nearby lighthouse is 16 degrees.
3. After traveling a further 50m directly away from the lighthouse, the angle decreases to 12 degrees.

Let's visualize this:

\ |
\ | 16°
\ |
\__|
. \
. \
. \
. \
.50m Lighthouse
.________

To find the height of the topmost point of the lighthouse above sea level, we'll need to use some trigonometric ratios.

Let's denote:
- h = height of the topmost point of the lighthouse above sea level.
- d = distance between the person and the lighthouse.

In this case, we are given the angle and the distance before and after traveling:

- Angle 1 (θ₁) = 16 degrees
- Angle 2 (θ₂) = 12 degrees
- Distance (d) = 50m

Using trigonometry, we can set up the following equation:

tan(θ) = opposite/adjacent

To find the height of the lighthouse (h), we know that:

tan(θ₁) = h/d

Solving for h, we get:

h = d * tan(θ₁)

Substituting the values we know:

h = 50m * tan(16 degrees)

Using a calculator, we can find tan(16 degrees) ≈ 0.2919. Substituting the value:

h = 50m * 0.2919

h ≈ 14.595 meters

Therefore, the height of the topmost point of the lighthouse above sea level is approximately 14.595 meters.