# Pilot

How high is the airplane's pilot to see 0.001 of Earth's surface?

Result

h =  12.782 km

#### Solution:

$R = 6378 \ km \ \\ S_{ 1 } = 4 \pi \cdot \ R^2 = 4 \cdot \ 3.1416 \cdot \ 6378^2 \doteq 511185932.522 \ km^2 \ \\ S = 0.001 \cdot \ S_{ 1 } = 0.001 \cdot \ 511185932.522 \doteq 511185.9325 \ km^2 \ \\ \ \\ S = 2 \pi R v \ \\ v = S / (2 \pi \cdot \ R) = 511185.9325 / (2 \cdot \ 3.1416 \cdot \ 6378) = \dfrac{ 3189 }{ 250 } = 12.756 \ km \ \\ y = R - v = 6378 - 12.756 = 6365.244 \ km \ \\ x = \sqrt{ R^2-y^2 } = \sqrt{ 6378^2-6365.244^2 } \doteq 403.1784 \ km \ \\ \ \\ x/y = (h+v)/x \ \\ h+v = x^2/y \ \\ h = x^2/y - v = 403.1784^2/6365.244 - 12.756 = \dfrac{ 6378 }{ 499 } \doteq 12.7816 = 12.782 \ \text { km }$

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#### Following knowledge from mathematics are needed to solve this word math problem:

Pythagorean theorem is the base for the right triangle calculator. See also our trigonometric triangle calculator.

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