Question

A pilot is traveling at a height of 30,000 feet above ground she looks down at a angle of depression of 6 and spots runway as measured along the ground how many miles away is she from the runway

Answer 1

Distance between runway and pilot position along the ground is 285430.9336 feet that is 53.9464 miles.

Solution:

Given that

Height of position of pilot from the ground = 30000 feet

Angle of depression when he looks down at runway = 6o

Need to measure along the ground, distance between runway and pilot that is horizontal distance between runway and pilot.

Consider the figure attached below

D represents position of runway.  

P represents position of pilot.

PG represents height of position of pilot from the ground that means PG = 30000 feet

PH is virtual horizontal line and HPD is angle of depression means ∠ HPD = 6 degree

AS DG and HP are horizontals, so DG is parallel to HP.

=>  ∠ HPD =∠ PDG =  6 degree  [ Alternate interior angle made by transversal PD of two parallel lines ]

We need to calculate DG

Consider right angles triangle PGD right angles at G

$\text {As } \tan x=\frac{\text { Perpendicular }}{\text { Base }}$

$\tan \angle \mathrm{PDG}=\frac{\mathrm{PG}}{\mathrm{GD}}$

$\begin{array}{l}{=>\mathrm{GD}=\frac{\mathrm{PG}}{\tan \angle \mathrm{PDG}}} \\\\ {=>\mathrm{GD}=\frac{30000}{\tan 6^{\circ}}=285430.9336}\end{array}$

As one foot = 0.000189 miles

$=>285430.9336 \text { feet }=285430.9336 \times 0.000189 \text { miles }=53.9464 \text { miles. }$

Hence distance between runway and pilot position along the ground is 285430.9336 feet that is 53.9464 miles.