On level ground, you stand 21.0m from the base of a tree and determine that the treetop is at 46 degrees above the horizontal, as measured from ground level.
How tall is the tree?
tan46 = h/21,
h = 21*tan46 = 21.7 m.
To determine the height of the tree, we can use trigonometry. We have the distance from the base of the tree (21.0m) and the angle (46 degrees) above the horizontal at which we're measuring the treetop. Drawing a diagram can help visualize the problem.
1. Identify the right triangle formed by the tree, the ground, and the line connecting your viewpoint to the treetop.
2. The height of the tree is the opposite side of the angle, while the distance from the base of the tree is the adjacent side.
3. We can use the tangent function to relate the angle and the sides of the triangle.
tan(angle) = opposite / adjacent
tan(46 degrees) = height / 21.0m
4. Rearrange the equation to solve for the height:
height = tan(46 degrees) * 21.0m
Now, we can calculate the height of the tree by plugging in the numbers and using a calculator:
height = tan(46 degrees) * 21.0m
height ≈ 22.37m
Therefore, the tree is approximately 22.37 meters tall.