A ship is 100 meters from a lighthouse. The angle of depression from the top of the lighthouse to the ship is 20°. What is the distance of the line of sight from the ship to the top of the lighthouse?

To solve this problem, we can use the tangent of the angle of depression. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.

Here, the angle of depression is 20°, the adjacent side (the distance from the lighthouse to the ship) is 100 meters, and the opposite side is the height of the lighthouse, which we are trying to find.

The formula for tangent is:

tan(θ) = opposite/adjacent

We can rearrange this formula to solve for the opposite side (the height of the lighthouse):

opposite = tan(θ) * adjacent

Substituting the given values:

height = tan(20°) * 100 meters

Using a calculator, tan(20°) is approximately 0.364, so:

height = 0.364 * 100 meters = 36.4 meters

Now, to find the line of sight from the ship to the top of the lighthouse, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the line of sight) is equal to the sum of the squares of the lengths of the other two sides (the height of the lighthouse and the distance from the lighthouse to the ship).

The formula for the Pythagorean theorem is:

a² + b² = c²

where a and b are the lengths of the two sides, and c is the length of the hypotenuse.

Substituting the given values:

(100 meters)² + (36.4 meters)² = c²

10000 meters² + 1324.96 meters² = c²

11324.96 meters² = c²

Taking the square root of both sides to solve for c:

c = sqrt(11324.96 meters²) = 106.42 meters

So, the distance of the line of sight from the ship to the top of the lighthouse is approximately 106.42 meters.