Two dogs are 180 ft apart on opposite sides of a tree. The angles of elevation from the dogs to the top of the tree are 35 degrees and 23 degrees.

Question

What is the height of the tree?

Answer 1

Let X be the distance of the dog with the 35 degree elevation angle from the tree. The other dog is 180-X away (in feet). Let H be the height of the tree. You have these two equations in two unknowns:
H/X = tan 35 = 0.7002075
H/(180-X) = tan 23 = 0.4244748
Solve for X first by eliminating H.
(180-X)/X = tan35/tan23 = 1.64959
180 - X = 1.64959 X
X = 180/2.64959 = 67.935 feet
180-X = 112.065 ft
H = X tan 35 = 47.569 ft

Answer 2

Thanks!