An observer, whose eyes are 1.97 m above the ground, is standing 47.0 m away from a tree. The ground is level, and the tree is growing perpendicular to it. The observer's line of sight with the treetop makes an angle of 25.0° above the horizontal. How tall is the tree?
1.97 + 47 tan 25
To find the height of the tree, we can use basic trigonometry. Let's break it down step by step:
Step 1: Identify the relevant angles and sides in the problem.
In this case, the angle of interest is the angle between the observer's line of sight and the horizontal (25.0°). The side we want to find is the height of the tree.
Step 2: Draw a diagram.
It's helpful to visualize the problem. Draw a horizontal line to represent the ground, a vertical line to represent the tree, and a diagonal line to represent the line of sight from the observer to the treetop.
Step 3: Define the trigonometric ratio that relates the angle and the sides.
Since we want to find the height of the tree, let's define the tangent ratio:
tangent(angle) = opposite/adjacent
In this case, the opposite side is the height of the tree, and the adjacent side is the distance from the observer to the tree.
Step 4: Apply the trigonometric ratio.
tangent(angle) = height of the tree / distance to the tree
tangent(25.0°) = height of the tree / 47.0 m
Step 5: Solve for the height of the tree.
To isolate the height of the tree, multiply both sides of the equation by the distance to the tree:
height of the tree = tangent(25.0°) * 47.0 m
Using a calculator, you can find that the tangent of 25.0° is approximately 0.4663:
height of the tree = 0.4663 * 47.0 m
height of the tree ≈ 21.8871 m
Therefore, the tree is approximately 21.8871 meters tall.