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In Sullivan's textbook I got this problem,but I cannot make the solution to this(Using Pythagorean theorem):

The Gibb’s Hill Lighthouse, Southampton, Bermuda, 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 Correctness of this information.The brochure further states that ships $40$ miles away can see the light and 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? (Use $3960$ as the radius of the earth).

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Specifically how radius of the earth is connected to this problem?

Additionally,what will be the solutions for ships and planes?

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    $\begingroup$ The earth is round. $\endgroup$ – Nosrati Feb 3 '17 at 13:59
  • $\begingroup$ I make it 23.3 miles, not 26, because $\sqrt{(3960\frac{362}{5280})^2-3960^2} \approx 23.3$. $\endgroup$ – TonyK Feb 3 '17 at 14:30
  • $\begingroup$ @TonyK I know that but what will be the solution for ship and planes $\endgroup$ – Noman Feb 3 '17 at 14:39
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    $\begingroup$ On a flat earth, you can't go past the horizon, you are always visible. $\endgroup$ – Yves Daoust Feb 3 '17 at 15:03
  • $\begingroup$ You knew that already? Good. Then you can work out the rest for yourself, can't you? $\endgroup$ – TonyK Feb 3 '17 at 18:04
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Having the radius of the earth helps us to find the distances of objects in horizon and beyond that. enter image description here

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  • $\begingroup$ what will be the height of the ship?and what about plane? $\endgroup$ – Noman Feb 3 '17 at 15:59
  • $\begingroup$ @Noman, I didn't include the calculation because you just asked for how the radius of the earth has connected to the problem. For the height of the ship, you have the arc length (the distance from lighthouse to the ship) 40 miles, you can easily find the angle between the lighthouse and the ship and also you have the radius of the earth. Now you have a right angled triangle with earth radius as one side and earth radius plus the ships height as its hypotenuse. $\endgroup$ – Seyed Feb 3 '17 at 16:43
  • $\begingroup$ i got it,you are right, i asked the question about posiibility $\endgroup$ – Noman Feb 3 '17 at 17:05
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The top of the lighthouse, the centre of the earth and a point at sea level from which the light is on the horizon form a right triangle.

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  • $\begingroup$ For observers above the surface of the sea (say, on a big ship or in an airplane) we have a triangle composed of two such right triangles. Their common right vertex is the tangency point of the light ray (lamp–observer line) and the sea surface. $\endgroup$ – CiaPan Feb 3 '17 at 14:11

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