From the deck of a ship, the light at the top of a light house is visible just above the horizon. The heights of the top of light house and the eye of the viewer from the ship above mean sea level may be assumed as 85 m and 6 m respectively. Assuming the radius of the earth as 6370 km and the usual correction refraction, determine the distance between the ship and light house.

Ship and Lighthouse Problem and Solution

Problem Statement

Solution

Define the variables for the problem:

Let A be the position of ship
Let B be the position of light house
Let O be the point where ray touches the sea
Let \(d_1\) be distance between ship and O
Let \(d_2\) be distance between lighthouse and O

Recall the formula for combined correction due to curvature and refraction:

\[h = \frac{6}{7} \times \frac{d^2}{2 \times 6370} \times 1000 \text{ meters}\] \[h = \frac{6}{7} \times \frac{d^2}{12740} \times 1000 \text{ meters}\] \[h = 0.06728 \times d^2 \text{ meters}\]

Rearranging to find distance d in terms of height h:

\[d = \sqrt{\frac{h}{0.06728}}\] \[d = 3.8553 \times \sqrt{h} \text{ kilometers}\]

Calculate the distance \(d_1\) from ship to point O using height of 6m:

\[d_1 = 3.8553 \times \sqrt{6}\] \[d_1 = 3.8553 \times 2.4495\] \[d_1 = 9.44 \text{ kilometers}\]

Calculate the distance \(d_2\) from lighthouse to point O using height of 85m:

\[d_2 = 3.8553 \times \sqrt{85}\] \[d_2 = 3.8553 \times 9.2195\] \[d_2 = 35.54 \text{ kilometers}\]

Calculate the total distance between ship and lighthouse:

\[\text{Total distance } AB = d_1 + d_2\] \[\text{Total distance } AB = 9.44 + 35.54\] \[\text{Total distance } AB = 44.98 \text{ kilometers}\]

Therefore, the distance between the ship and lighthouse = 44.98 kilometers

Detailed Explanation

This problem involves finding the maximum distance from which an observer can see an object, considering Earth’s curvature and atmospheric refraction.

The key insight is that when the lighthouse light is “just visible above the horizon” from the ship, the line of sight is tangent to the Earth’s surface at point O. This creates two separate visibility distances that add together:

\(d_1\): The distance from the ship to point O – This depends on the height of the observer’s eye above sea level (6m)
\(d_2\): The distance from the lighthouse to point O – This depends on the height of the lighthouse (85m)

The formula used (\(h = 0.06728 \times d^2\)) is derived from geometric principles involving the radius of the Earth (6370 km) and includes a factor (6/7) that accounts for atmospheric refraction. The formula relates the height above sea level to the maximum distance visible.

By solving for distance in terms of height (\(d = 3.8553 \times \sqrt{h}\)), we can calculate how far an observer at a specific height can see to the horizon (\(d_1\)) and how far the lighthouse can be seen from the horizon (\(d_2\)).

The total distance is the sum of these two components, which gives us the maximum distance at which the lighthouse is visible from the ship: 44.98 kilometers.

Problem Statement

Solution

Detailed Explanation

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