From a lookout tower 80 ft. high, a man observes from a position 6.5 ft. below the top of the tower that the angle of elevation of the top of a certain tree is 12 degrees 40 minutes and the angle of depression of its base is 72 degrees 20 minutes. If the base of the tower and the base of the tree are at the same level, what is the height of the tree?
If you draw a diagram, you will see that you cannot compute the height of the tree without knowing how far apart are the tower and the tree.
The observer is 80-6.5=73.5 above the base of the tower and the base of the tree.
An angle of depression to the base of the tree of 72-20-00 means that the distance between the tower and the tree is D=73.5/tan(72-20-00)
the difference between the top of the tree and the observer is therefore
h=D sin(12-40)
The height of the tree, H, is then
H=73.5+D sin(12-40)
=73.5 + 73.5 sin(12-40-00)/tan(72-20-00)
=73.5(1+sin(12-40-00)/tan(72-20-00))
=78.6 ft
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To find the height of the tree, we need to use trigonometry. Let's break down the problem step by step:
Step 1: Draw a diagram
First, draw a diagram to visualize the problem. Label the lookout tower as A, the tree as B, and the base of the tree as C.
A (lookout tower)
----------------
\ |
\ |
\ | h (height of tree)
\|
B (top of tree)
|
|
|
|
C (base of tree)
Step 2: Identify the given information
We are given the height of the lookout tower (80 ft) and the angle of elevation to the top of the tree (12 degrees 40 minutes) from the man's position 6.5 ft below the top of the lookout tower. We are also given the angle of depression to the base of the tree (72 degrees 20 minutes).
Step 3: Find the height of the tree
To find the height of the tree (h), we can use trigonometric ratios. Since we have the angle of elevation and the adjacent side of the triangle (6.5 ft), we can use the tangent function:
tan(angle) = opposite / adjacent
tan(12 degrees 40 minutes) = h / 6.5 ft
Since angles are typically measured in degrees, we need to convert the angle to degrees and decimal format:
12 degrees 40 minutes = 12 + (40/60) = 12.67 degrees
Using the tangent formula:
tan(12.67 degrees) = h / 6.5 ft
Now, let's solve for h:
h = tan(12.67 degrees) * 6.5 ft
Using a scientific calculator:
h ≈ 1.4332 * 6.5 ft
h ≈ 9.3178 ft
Therefore, the height of the tree is approximately 9.3178 ft.