Question

You have climbed to the top of a tall tree. When you get to the top, you use your clinometer to discover that the angle between the tree and the line of sight to your red lunchbox is 30°. You know you left the lunchbox 20 meters from the base of the tree. How tall is the tree?

A. 75.36 m
B. 92.09 m
C. 20.17 m
D. 51.25 m
E. 18.95 m
F. 34.64 m

Answer 1

F. 34.64 m

Explanation:

The measurements given in the question are;

The angle (of depression) given by the clinometer, θ = 30°

The horizontal distance of the lunchbox from the tree, d = 20 meters

The height (how tall) of the tree = Required

Let the height of the tree be assumed to be perpendicular to the ground, noting that the horizontal distance from the lunchbox to the tree is a straight line, let l represent the line of sight from the top of the tree to the lunchbox, and let h represent the height of the tree we have;

The line of sight to the lunchbox, l, the height of the tree, h, and the horizontal distance of the lunchbox from the base of the tree form a right triangle

The height of the tree is the adjacent leg to the given angle by the clinometer

Using trigonometric ratios, we have;

tan(30°) = d/h

∴ tan(30°) = (20 m)/h

h = (20 m)/tan(30°) ≈ 34.64m

The height of the tree, h ≈ 36.64 m.