Question

You want to measure the height of a tree growing vertically on the side of a hill. The hill makes an angle of 20°with the horizontal. Standing 15ft. downhill from the base of the tree, you measure the angle formed by the hill and the top of the tree to be 45°. Find the height of the tree to the nearest foot. Show your work.

Answer 1

We need to solve for the height of the tree and we need to define the variable that would represent this value. In this case, we assign "y" for the unknown height of the tree. See attached picture for better comprehension.
Solving for the variable "a" which is base of downhill to the tree, we have it:
cos 20° = a / 15 ft
a = 15*cos 20°
a = 14.095 feet
Solving for the variable "x" which is the side of the hill to the base of the tree, we have it:
sin 20° = x / 15 ft
x = 15 * sin20°
x = 5.130 feet
Solving for the variable "y", which is the height of the tree, we have it:
tan 45° = (x+y) / a
tan 45° = (5.13 + y) / 14.095
y = 14.095*tan45° - 5.13
y = 8.965 feet

The height of the tree is 8.965 feet.