Question

Two observers on opposite sides are 350m apart and are looking at a balloon if the angles of elevation from each observer are 30degree and 60degree respectively,how high is the balloon?

Answer 1

make your sketch
let the point below the ballon be P
let the distance from the 60° position to P be x
then the other distance is 350-x

You have two right-angled triangles
let the height be h m (the common side to the two triangles)

for the 1st triangle:
tan60 = h/x
h = xtan60

for the 2nd:
tan30 = h/(350-x)
h = (350-x)tan30
so
xtan60 = (350-x)tan30
xtan60 = 350tan30 - xtan30
xtan60 + xtan30 = 350tan30
x(tan60 + tan30) = 350tan30
x = 350tan30/(tan60+tan30)

back to the 1st triangle
h = xtan60

h = 350tan30tan60/(tan60+tan30)
= ...
Only now would I reach for my calculator and do all the calculations in one swoop using the ( ) keys to make sure of my order of operations.

I got appr 151.55 m

Answer 2

thank you sir it is really helpful :)