Question 469718
Draw 3 lines from the woman to the flag pole , one to the top, one to the bottom and one straight across perpendicular to flag pole.
This forms 2 right triangles.
Angle opposite flag pole in top triangle is 18 degrees
Angle opposite flag pole in bottom triangle is 14 degrees
The adjacent side for both angles is the distance from the flag pole, call it d.
Let the length of the top part of flag pole be x, then length of bottom part is 36-x.
In this way their sum equals 36, the length of the flag pole.
Use trig relationships to solve for d
Note that tan(theta) = opp/adj
Thus
{{{tan(18) = x/d}}}
{{{tan(14) = (36-x)/d}}}
Solve for x in 1st equation:
{{{x = d*tan(18)}}}
Substitute this in for x in 2nd equation
{{{tan(14) = (36 - d*tan(18))/d}}}
Solve for d:
Multiply by d on both sides
{{{d*tan(14) = 36 - d*tan(18)}}}
Add d*tan(18) on both sides
{{{d*tan(14) + d*tan(18) = 36}}}
Factor d from left side
{{{d(tan(14)+tan(18)) = 36}}}
Divide by tan(14)+tan(18) on both sides
{{{d = 36/(tan(14)+tan(18))}}}
Using scientific calculator
{{{d = 62.69}}}
Therefore the woman is approximately 62.7 ft away from the flag pole.