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Earth is roughly 8000 miles in diameter.

(a) I'm riding in a hot air balloon 1 mile above the surface of earth. Approximately how far away is the horizon? (In other words, how far away is the farthest point on the surface of Earth that I can see.)

(b) What if I'm in a plane 6 miles above the surface of the earth?

(c) What is I'm in a spaceship 100 miles above the surface of the earth?

Mathgenius Oct 6, 2018

Aug 14, 2020

#1
+1

We  can  figure the distance  to the horizon  using a right triangle

The distance from  the center of the Earth to any object above the Earth's surface  = the hypotenuse = 8000  + h      where h is the height of the object [in  miles ] above the Earth's surface

And the radius of the Earth  is equal to one leg of the right triangle

And the distance from the object to the horizon, D,  is the other leg

So...we have this general formula to determine D

D  = √ [ (8000 + h)^2  - 8000^2]

So....

(a)  D  = √ [ (8000 + 1)^2  - 8000^2 ]  ≈ 126.5 miles

(b)  D  =   √ [ (8000 + 6)^2  - 8000^2 ]  ≈  309.9  miles

(c)  D  = √ [ (8000 + 100)^2  - 8000^2 ]  ≈  1268.9 miles

Aug 14, 2020
#3
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Guest #1

Guest Aug 14, 2020
#2
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There is a very simple and quite accurate formula that can be used to meassure the distance to the Horizon:

D =Sqrt[1.5  x  H],  where H = Height in feet and D = The distance in Miles.

a - Sqrt[1.5 x 5,280] =~89 miles

b - Sqrt[1.5 x  31,680] =~218 miles.

c - Sqrt[1.5 x  528,000] =~890 miles.

Aug 14, 2020
#4
+112062
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Here is the logic behind it.

Just use Pythagoras's theorem to solve it.

Aug 15, 2020