Science at the beach.

I visited Kingsbarns beach in Fife and a distant object caught my eye. It was the Inchcape (Bell Rock) lighthouse, and it could be seen clearly from were I parked my car at around 6 metres above the beach. I took this picture using my phone at 30x zoom.

The full lighthouse can be seen on this link: lighthouseaccommodation.co.uk/listings/bell-rock-lighthouse

The lighthouse was so far out I wasn’t seeing all of it. Then when I went down to the beach, I could see only the top sticking up (see the PIP). A difference in height of 4-5 metres meant I was seeing a lot less of the tower.

So, I questioned Grok about this. Grok did the following calculation based on the above picture, knowing the lighthouse was 36m tall.
Formula: 3.57 × (√h₁ + √h₂)
• Your eye height on the bank: ≈ 6 m
• Visible height of lighthouse in the first photo: roughly the top 15–18 m (out of the full 36 m tower)
So √6 ≈ 2.45, √17 ≈ 4.1 → sum ≈ 6.55 3.57 × 6.55 ≈ 23.4 km.

So, from this picture, Grok had calculated a distance of 23.4km. I have checked the distance using OS maps; the distance was just short of 22km. I was impressed and these pictures show that view of the lighthouse, from this distant vantage point, is affected by the curve of the earth. There is no other explanation.