Finding the Distance Between Two People on a Curved Earth

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SUMMARY

The discussion focuses on calculating the distance two people, each 1.8 meters tall, must walk away from each other until they can no longer see one another due to the Earth's curvature, with a radius of approximately 6378 km. The solution involves using trigonometric principles to establish a right triangle where the radius of the Earth and the height of the individuals are key components. The angle subtended at the center of the Earth, denoted as θ, is crucial for determining the arc length, which is calculated using the formula rθ, where r is the radius of the Earth plus the height of the individuals. This arc length represents half the distance the individuals must walk apart.

PREREQUISITES
  • Understanding of basic trigonometry, specifically right triangles
  • Familiarity with the concept of arc length in circular geometry
  • Knowledge of the Earth's curvature and its radius (approximately 6378 km)
  • Ability to visualize geometric relationships in two-dimensional diagrams
NEXT STEPS
  • Study the derivation of the arc length formula in circular geometry
  • Learn about the properties of tangents to circles and their relationship with radii
  • Explore applications of trigonometry in real-world scenarios involving curvature
  • Investigate the effects of height on visibility over large distances on a spherical surface
USEFUL FOR

Students studying geometry and trigonometry, educators teaching mathematical concepts related to curvature, and anyone interested in practical applications of mathematics in real-world scenarios involving distance and visibility on Earth.

Venito
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I have not done any maths for a very long time now. And have forgotten a lot of what I learned. Right now I am having to get back into it again. And can't get my head around this problem.


Homework Statement



Two people 1.8m tall walk directly away from each other until they can no longer see each other. [Due to the curvature of the Earth radius of about 6378km]

The part of this question I don't get which I am trying to find is how to get this part.


Assuming nothing else blocks their view, how far do they have to walk to not see each other? [Hint look for the length of the arc.]

Homework Equations



Dont know how to show my working out. I am in the diagram stage. And trying to get a formula. Just can't remember fully what I am doing.



The Attempt at a Solution



For a attempt, well I get know were. And my txt books are no help. All talking about it in graph format.
 
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Ok, in your diagram you can simplify things by noticing the symmetry of the landscape between each person. You should draw up one man, the curvature of the Earth, and his line of view from the top of his head just brushing passed the Earth (just as he's about to lose sight of the other man).

You will need to include the centre of the Earth and create a radius to the man and to the point where the line of sight touches the Earth. This line of sight will be tangent to the circle (Earth) and remember that tangents to circles are perpendicular to the radius. That is, you're now dealing with a right-triangle.
Let [itex]\theta[/itex] be the angle subtended at the centre of the Earth, r (radius of Earth) be the adjacent side, and r+1.8 (since this is the height of the man and the radius of the Earth) be the hypotenuse. Now using trigonometry, you can find [itex]\theta[/itex] in terms of the radius and the height of the man.

Lastly, there is a rule that says the length of an arc is [itex]r\theta[/itex]. But remember that the length of the arc in this case is half the length of the actual distance required for the men to walk away from each other (since we're dealing with only one symmetrical half of the problem).

Can you take it from here?
 

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