A person standing 50 m away from a tree observes the angle of elevation to the t

A person standing 50 m away from a tree observes the angle of elevation to the top of the tree as 30 degrees. What is the height of the tree?

A person standing 50 m away from a tree observes the angle of elevation to the top of the tree as 30 degrees. What is the height of the tree?

Step 1: Understand the problem. A person is standing 50 meters away from a tree and sees the top of the tree at an angle of 30 degrees above the horizontal.
Step 2: Identify the right triangle formed by the person, the top of the tree, and the base of the tree. The distance from the person to the tree is the base, and the height of the tree is the vertical side.
Step 3: Use the tangent function, which relates the angle of elevation to the opposite side (height of the tree) and the adjacent side (distance from the tree). The formula is: tan(angle) = opposite/adjacent.
Step 4: Substitute the known values into the formula. Here, angle = 30 degrees, opposite = height of the tree, and adjacent = 50 meters. So, tan(30°) = height / 50.
Step 5: Calculate tan(30°). The value of tan(30°) is 1/√3.
Step 6: Rewrite the equation: 1/√3 = height / 50.
Step 7: Solve for height by multiplying both sides by 50: height = 50 * (1/√3).
Step 8: Calculate the height: height = 50 / √3, which is approximately 25 meters.

Trigonometry – The problem involves using the tangent function to relate the angle of elevation to the height of the tree and the distance from the observer.
Angle of Elevation – Understanding how to interpret the angle of elevation in relation to a right triangle formed by the observer, the top of the tree, and the base of the tree.
Right Triangle Properties – Applying properties of right triangles to solve for unknown lengths using trigonometric ratios.