Question

A forest ranger stood at a point that formed a 20° angle with the top of a tree and the ground, as shown in the diagram below.

A right triangle with height h, base x and base angle is 20 degrees.

Which equation can be used to find
, the height of the tree, in terms of
, the distance from the ranger to the tree?

Answer 1

To find the height of the tree (\(h\)) in terms of the distance from the ranger to the tree (\(x\)), you can use the tangent function from trigonometry.

In a right triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. In this case:

- The height of the tree (\(h\)) is the length of the opposite side.
- The distance from the ranger to the base of the tree (\(x\)) is the length of the adjacent side.
- The angle between the ground and the line of sight to the top of the tree is 20°.

The tangent of the angle (20°) can be expressed as:

\[
\tan(20°) = \frac{h}{x}
\]

To solve for the height of the tree (\(h\)), you can rearrange this equation:

\[
h = x \cdot \tan(20°)
\]

Thus, the equation that can be used to find the height of the tree in terms of the distance from the ranger to the tree is:

\[
h = x \tan(20°)
\]