An observer, whose eyes are 1.87 m above the ground, is standing 27.0 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 18.0° above the horizontal. How tall is the tree?
1.87m + 27.0m tan(18.0°)
To find the height of the tree, we can use trigonometry. Let's break down the problem step by step.
Step 1: Draw a diagram.
Visualize the observer, the tree, and the horizontal ground. Label the height of the observer above the ground as "h" and the distance between the observer and the tree as "d". Also, draw a line segment from the treetop to the ground, representing the height of the tree, and label it as "x" (what we want to find).
Step 2: Identify the relevant trigonometric values.
In this problem, the angle of elevation (the angle between the observer's line of sight and the horizontal ground) is given as 18.0°. We can apply trigonometry to the right triangle formed by the observer, the tree, and the line segment from the treetop to the ground.
Step 3: Apply trigonometric functions.
Since we are given the angle of elevation and are trying to find the height of the tree, we will use the tangent function (tan).
The tangent function is defined as tan(θ) = opposite / adjacent.
In this case, the opposite side is the height of the observer (h) plus the height of the tree (x), and the adjacent side is the distance between the observer and the tree (d).
Therefore, we can write the equation as:
tan(18.0°) = (h + x) / d
Step 4: Solve for x (height of the tree).
We rearrange the equation to solve for x:
x = d * tan(18.0°) - h
Step 5: Substitute the given values and calculate.
Now, plug in the known values:
d = 27.0 m
h = 1.87 m
θ = 18.0°
x = 27.0 * tan(18.0°) - 1.87
Using a calculator, calculate the value of tan(18.0°), and then substitute it into the equation to find x (height of the tree).
x ≈ 9.23 m
Therefore, the height of the tree is approximately 9.23 meters.