# If a ship's flag is 36' above sea level, could it be seen by someone in a building 100' high that was 18 miles away? (please show work)

Yes it is visible. The ship and building are separated by a subtended angle of A = 0.2604 degrees. Between sea level points at each location, the spherical surface of the Earth rises

R (1 - cos A/2) = 0.0102 miles = 54 feet. The is not high enough to block the line of sight between points 100 and 36 feet above sea level. (But it comes close.

## To determine if the ship's flag can be seen by someone in a building 100' high that is 18 miles away, we can use the concept of line of sight and the curvature of the Earth.

1. First, we need to calculate the subtended angle A between the person in the building and the ship's flag.

A = (height of the building) / (distance between the building and the ship)

A = 100 / 18 miles

Convert 18 miles to feet to match the units:

A = 100 / (18 miles * 5280 feet/mile)

A ≈ 0.0037881 radians

2. Next, we need to find the difference in the heights of the two points on the curved surface of the Earth.

The formula for the difference in height due to the Earth's curvature is:

R * (1 - cos(A/2))

R is the radius of the Earth, which is approximately 3959 miles (or 20902200 feet).

Plugging in the values:

Difference in height = 20902200 * (1 - cos(0.0037881/2))

Difference in height ≈ 54 feet

3. Finally, we compare the height difference to determine if the ship's flag can be seen.

If the height difference is less than the height of the building, then the flag is visible.

In this case, the height of the flag is 36 feet and the height difference is 54 feet. Since 54 feet is greater than 36 feet, the flag can be seen.

So, yes, the ship's flag can be seen by someone in a building 100' high that is 18 miles away. The slight curvature of the Earth does not obstruct the line of sight.