Question

In operation since 1846 , stands 117 feet high on a hill 245 feet high, so its beam of light is 362 feet above sea level. A brochure states that the light itself can be seen on the horizon about 26 miles distant. Verify the accuracy of this information.The brochure further states that ships 40 miles away can see the light and that planes flying at 10,000 feet can see it 120 miles away. Verify the accuracy of these statements. What assumption did the brochure make about the height of the ship?

Short answer

The light can be seen up to 23.3 miles away, not 26. For a ship 40 miles away to see it, the ship must be 1066 feet tall. Planes at 10,000 feet can see it from 120 miles.

Step-by-step solution

- Calculate the Distance to the Horizon for the Lighthouse

The distance to the horizon from a height is given by the formula \[ d = \sqrt{2hR} \] where \( h \) is the height above sea level and \( R \) is the radius of the Earth. Here, \( R = 3960 \) miles and \( h = 362 \) feet or approximately 0.0686 miles. Thus, \[ d = \sqrt{2 \cdot 0.0686 \cdot 3960} = \sqrt{543.12} \approx 23.3 \text{ miles} \] as the distance to the horizon.

- Verify Light Can Be Seen 26 Miles Away

To verify if the light can be seen 26 miles away, why need to add the horizon distances for both the lighthouse and the observer at sea level: \[ d = \text{Horizon distance for lighthouse} + \text{Horizon distance for observer} \] Since the observer is at sea level, their horizon distance is 0. Thus, the total visible distance from the lighthouse is approximately 23.3 miles, less than the claimed 26 miles.

- Verify Visibility for Ships 40 Miles Away

For a ship 40 miles away to see the lighthouse, the ship must be able to see beyond its horizon distance. Using the same horizon formula, calculate the required height of the ship: \[ d = \sqrt{2hR} \implies 40 = \sqrt{2h \cdot 3960} \] Solving for \( h \), we get: \[ 40^2 = 2h \cdot 3960 \implies h \approx 0.202 \text{ miles} \approx 1066 \text{ feet height} \] This suggests the ship must be exceptionally tall to see the lighthouse 40 miles away.

- Verify Visibility for Planes Flying at 10,000 Feet

Calculate the horizon distance from a plane flying at 10,000 feet (or approximately 1.8939 miles): \[ d = \sqrt{2 \cdot 1.8939 \cdot 3960} \approx \sqrt{14984.8} \approx 122 \text{ miles} \] This verifies the statement that a plane flying at 10,000 feet can see the lighthouse from 120 miles away.

- Assumption about Ship Height

The brochure assumes a ship height of around 1066 feet to see the lighthouse from 40 miles away, which is highly unrealistic. Thus, the claim is inaccurate unless the ship were exceptionally tall.

Key concepts

Lighthouse Height

Understanding the height of a lighthouse is crucial to calculating visibility distances. In this exercise, the lighthouse stands on top of a hill and has a combined height above sea level of 362 feet. This height determines how far its light can reach over the curvature of the Earth. Lighthouses are built tall to maximize their visibility to ships at sea. When we know the height, we can apply the distance to horizon formula to find how far away the light can be seen.

Distance to Horizon Formula

The formula to calculate the distance to the horizon is \( d = \sqrt{2hR} \), where \( h \) is the height above sea level and \( R \) is the Earth’s radius. In this exercise, \( h \) is 362 feet (about 0.0686 miles) and \( R \) is 3960 miles. Plugging these values into the formula, we get \[ d = \sqrt{2 \cdot 0.0686 \cdot 3960} = \sqrt{543.12} \approx 23.3 \text{ miles} \]. This calculation shows how far the lighthouse’s light can reach over the Earth's curvature.

Earth's Radius Effect

The Earth's radius is a critical factor in determining visibility. Since the Earth is a sphere, the curvature limits how far we can see based on height. Using the radius of 3960 miles, we see that higher altitudes allow us to see further distances. For instance, the horizon distance significantly increases when viewed from planes at 10,000 feet due to the larger value of \( h \). This spherical effect means that visibility calculations must account for both the observer's height and the object's height to be accurate.

Visibility Limits

Visibility limits from objects like lighthouses depend on both the height of the object and the observer. For a ship 40 miles away to see the lighthouse, the ship must have a significant mast height, calculated to be around 1066 feet, which is unrealistic for typical ships. This reveals that there are practical limits to visibility based on realistic heights of ships and observers. Planes flying at higher altitudes can see further because they overcome more of the Earth’s curvature. The horizon formula helps understand these limits and confirm the brochure's claims about visibility distances.