trigonometry
posted by carlo .
howard and keight who are 2km apart are on a horizontal plane observe a balloon in the same vertical plane with themselves the angles of elevation are 50 degrees and 65 degrees respectively find the height of the balloon.
A.if its situated between howard and keight
B.If it is on the side of both of them.
Help plz

In your sketch, draw an altitude hitting the base line at P, Howard to P is HP, Keight to P is KP
Let HP = x ,angle at H is 50°
then KP = 2x, angle at K is 65
let the height be h
I see two right  angled triangles,
in the first: cot50 = x/h
x = hcot50
in the second: cot 65 = (2x)/h
hcot65 = 2x
x = 2  hcot65
so
hcot50 = 2hcot65
hcot50 + hcot65 = 2
h(cot50+cot65) = 2
h = 2/(cot50+cot65)
(only now do we have to go to a calculator, I will let you do the buttonpushing)
for the second part, I see it as
cot50 = (2+x)/h and cot 65 = x/h
proceed the same way as above.