Question

A friend is visiting a very tall tree. She thinks it might actually be the tallest tree in the world, or close to it, and she wants to find out how tall it is. She lays down 80 feet from the tree and looks up at the very top of the tree at 75˚. How tall is the tree?.

Answer 1

By using trigonometry, specifically the tangent function, and knowing the distance to the tree (80 feet) and the angle of elevation (75°), we can calculate that the height of the tree is approximately 298.56 feet.

Step-by-step explanation:

To determine the height of a tree based on an observation from a distance, one can employ trigonometry, specifically the tangent function, which relates the angle of elevation to the height and distance from the object. Here, the student's friend lies 80 feet from the tree and looks up at a 75° angle to see the top of the tree.

The tangent of the angle (75°) is equal to the opposite side (the height of the tree) divided by the adjacent side (the distance from the tree), in this case, 80 feet. So, we can set up the equation tan(75°) = height / 80 feet. By multiplying both sides by 80 feet, we can solve for the height of the tree.

Mathematically, this looks like:

height = 80 feet * tan(75°)

Using a calculator:

height = 80 feet * 3.732 (rounded tangent of 75°)

height = 298.56 feet

Therefore, the height of the tree is approximately 298.56 feet.