Question 1110603
A sketch representing the situation is shown below.
{{{drawing(400,250,-100,1900,-200,1100,
red(triangle(0,0,1732,0,1732,1000)),
green(triangle(1155,0,1732,0,1732,1000)),
green(rectangle(1732,0,1702,30)),
line(0,0,1155,0),arrow(1155,0,2100,0),
line(0,-5,1155,-5),locate(1700,1090,plane),
arrow(0,0,-200,0),locate(570,90,runway),
locate(1750,500,h),locate(1440,90,y),
locate(850,-25,x),arrow(830,-70,0,-70),
arrow(902,-70,1732,-70),locate(200,110,red(A)),
red(arc(0,0,400,400,-30,0)),arrow(1732,1000,600,1000),
red(arc(1732,1000,400,400,150,180)),locate(1550,1000,red(A)),
green(arc(1155,0,500,500,-60,0)),locate(1330,120,green(B)),
green(arc(1732,1000,500,500,120,180)),locate(1450,960,green(B))
)}}} For the a better illustration,
in the sketch the angles of depression are {{{30^o}}} and {{{60^o}}} .
{{{runway=x-y}}} ,
{{{tan(A)=h/x}}} --> {{{x=h/tan(A)}}} ,
{{{tan(B)=h/y}}} --> {{{y=h/tan(B)}}} .
With all length in feet,
{{{h=4000}}} , {{{A=6^o}}}{{{"40 '"=(6&2/3)^o=about 6.666667^o}}} , and {{{B=10^o}}} ,
we can calculate a good approximate value of the runway length as
{{{4000/tan(6.666667^o)}}}{{{"-"}}}{{{4000/tan(10^o)}}}{{{"="}}}{{{4000/0.116883}}}{{{"-"}}}{{{4000/0.176327}}}{{{"="}}}{{{34222-22685}}}{{{"="}}}{{{"about 11,500"}}} .
The length of the runway is about {{{highlight("11,500 feet")}}} .
We could carry more digits through the calculations,
and we could calculate the length of the runway as {{{11537feet}}} ,
but considering the uncertainty of the altitude and angle measurements
a more precise result is not more meaningful.