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# Geometry On Earth

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A pilot is flying a plane 4.5 mi above the earth’s surface.

From the pilot’s viewpoint, what is the distance to the horizon?

https://static.k12.com/nextgen_media/assets/1576278-IM3_150120_060404.jpg

Apr 12, 2018

#1
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a tangent and radius meet at 90 degrees

use Pythagorean theorem

c= 3959mi + 4.5mi

c= 3963.5mi

a^2 = c^2 - b^2

a^2 = 3963.5^2 - 3959^2

a^2 = 35651.25

a= √35651.25

a= 188.8mi (nearest tenth)

Apr 12, 2018
#2
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nevermind

Apr 12, 2018
edited by summertastic1011  Apr 12, 2018
#3
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0

Apr 12, 2018
#4
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no problem

Apr 12, 2018
#5
+557
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Apr 12, 2018
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A person standing at the top of Mountain Rainier would be approximately 2.7 mi high. The radius of earth is 3959 mi.

What is the distance to the horizon from this point?

Apr 12, 2018
#7
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nevermind

Apr 12, 2018
edited by lynx7  Apr 12, 2018
#8
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oh ok lol

Apr 12, 2018
edited by summertastic1011  Apr 12, 2018
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do we need to find the point at the centre of the earth or the point on the circumfernce of earth?

Apr 12, 2018
#10
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im not sure i dont think so

Apr 12, 2018
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Apr 12, 2018
#12
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oh ok how did you get the answe?

Apr 12, 2018
#13
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The line of sight to the horizon , the radius at the horizon and the radius at the mountain from a right angle triangle, so Pythagoras's theorem gives

d = sqrt( ((R+h)^2 - R^2)

= sqrt( 2Rh + h^2)

= 146.2mi

Apr 12, 2018
#14
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A person standing at the top of Mountain Rainier would be approximately 2.7 mi high. The radius of earth is 3959 mi.

What is the distance to the horizon from this point?

We can use the Pythagorean Theorem, here

We have a right triangle with a hypotenuse of 3959 + 2.7  = 3961.7 mi

And the radius of the Earth represents one of the legs...so....the distance to the horizon is the other leg and given by :

√ [ 3961.7^2  - 3959^2 ]   ≈  146.2 mi

Apr 12, 2018
#15
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thank you both so much!

Apr 12, 2018
#16
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no problem any time

Apr 12, 2018
#17
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There is a very simple formula for this kind of problem:

D = ~Sqrt[1.5 x H], where D =distance in miles, H=Height in feet.

D =~sqrt[1.5 x 2.7 miles x 5,280 feet]

D =~Sqrt[ 21,384]

D=~ 146.23 miles - distance to the horizon.

Apr 13, 2018