Two people standing 30 feet apart look east at the top of a tree. The first person looks up at an angle of 78 degrees. The second person looks up at an angle of 62 degrees. How tall is the tree (to the nearest foot)?

Answer 1

Refer to the diagram:
http://img3.imageshack.us/i/11260110467.png/

H = height of the tree
x = distance of tree from first observer in feet
x+30 = distance of tree from second observer in feet
h=average height of observers' eys from ground in feet.

Using the trigonometric ratio
tan(θ)=opposite/adjacent, we have

tan(78°)=H/x
H=x*tan(78°).....(1)
tan(62°)=H/(x+30)
H=(x+30)*tan(62°).....(2)

Equate H from (1) and (2)
x*tan(78°) = (x+30)*tan(62°)
Solve for x.

Substitute the value of x into (1) to get H.

Finally, add 5 feet to H for the average height of eyes above ground.

I get 99 feet including adjustment for distance from ground to observers' eyes.... quite a tree!

Answer 2

Did you make a diagram?
I labeled the two people as P and Q, with Q closer to the tree.
I labeled the top of the tree T and its base O
It is easy to see that angle PTO = 16º

by sine law:
QT/sin62º = 30/sin16º
So you can find QT, which is the hypotenuse of triangle TOQ

then sin 78º = TO/QT
TO = QTsin78º

evaluate.