Question

A vertical tree is growing on the side of a hill with gradient 10± to the horizontal. From a point 50 m downhill from the tree, the angle of elevation to the top of the tree is 18º. Find the height of the tree.

Answer 1

For this item, we think of a right triangle with hypotenuse equal to 50 m. Initially, the angle near this angle is equal to 10°. The height of the hill and the horizontal distance from the root of the tree to the point are calculated below,
                         height of hill, h:  sin 15° = h / 50   ; h = 12.94 m
                         horizontal distance, x:   cos 15° = x / 50   ; x = 48.30 m
Then, we solve for the vertical distance from the foot of the hill to the top of the tree by the given angle of elevation. 
                         tan 18° = y / 48.30 m    ; y = 15.69 m
We subtract the height of the hill from the value of y in order to determine the height of the tree,
                                   height of the tree = 15.69 m - 12.94 m = 2.75 m
Therefore, the height of the tree is equal to 2.75 m.