Asked by Ryan
On the bridge of a ship at sea, the captain asked the new, young officer standing next to him to determine the distance to the horizon. The officer took pencil and paper, and in a few moments came up with an answer. On the paper he had written the formula d = 3.6 square root h. Show that this formula is a good approximation of the distance, in kilometers, to the horizon, if h is the height, in meters, of the observer above the water. (Assume the radius of the earth to be 6500 km.) If the bridge was 27m. above the water, what was the distance to the horizon?
I understand how to find the distance by just plugging it into the formula, but I would like to know the deviation of the formula. Here is a picture of the problem: gyazo(DOT)com/2acde0ce3958354b207df75329356b8a
I understand how to find the distance by just plugging it into the formula, but I would like to know the deviation of the formula. Here is a picture of the problem: gyazo(DOT)com/2acde0ce3958354b207df75329356b8a
Answers
Answered by
Steve
no deviation to that formula. Maybe you'd like the derivation...
Draw a radius r, extended by h.
Draw a second radius r.
Connect the two endpoints, and that is the distance to the horizon.
d = √((r+h)^2-r^2) = √(2rh+h^2)
That's the distance in km. Usually h, in meters is very much smaller than r, so
d ≈ √(2rh)
Now, using km for everything,
d ≈ √(2*6500*h/1000) = √(13h) = 3.6√h
Draw a radius r, extended by h.
Draw a second radius r.
Connect the two endpoints, and that is the distance to the horizon.
d = √((r+h)^2-r^2) = √(2rh+h^2)
That's the distance in km. Usually h, in meters is very much smaller than r, so
d ≈ √(2rh)
Now, using km for everything,
d ≈ √(2*6500*h/1000) = √(13h) = 3.6√h
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