Question

Kate and John both are 100 m far away from each other. They both are 1.5 m tall. They flagpole are somewhere between them. Kate looking up the flagpole at the angle of elevation of 25 degree while John looking at 60 degree. How far the flagpole they both standing and how tall the flagpole?

Answer 1

Consider the height h of the pole above their eyeballs.

Let Kate be x meters from the pole

h/x = tan 25°
h/(100-x) = tan 60°

x tan 25° = (100-x)tan 60°
.4663x = 173.205 - 1.7321x
2.1984x = 173.205
x = 78.79

So, Kate is 78.79m from the pole and
John is 21.21m from the pole.

h = 78.79 tan 25° = 36.74

so, the pole is 36.74 + 1.5 = 38.24m high (assuming their eyeballs are on top of their heads...)