Answer: We can use basic trigonometry to solve both of these problems:
Angle of depression from the top of the tree to the lake:
Let's draw a diagram of the situation:
A (top of tree)
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30 | \ 15 ft
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|__________\
B (lake)
In this diagram, A represents the top of the tree, B represents the lake, and the lines connecting them form a right triangle. We want to find the angle of depression, which is the angle between the horizontal line (from the top of the tree to the lake) and the line of sight from the top of the tree to the lake. This is angle θ in the diagram.
We know that the opposite side of this right triangle is 30 feet (the horizontal distance from the top of the tree to the lake) and the adjacent side is 15 feet (the height of the tree). Therefore:
- tan(θ) = opposite/adjacent = 30/15 = 2
Taking the arctangent of both sides gives us:
- θ = arctan(2) ≈ 63.4 degrees
Therefore, the angle of depression from the top of the tree to the lake is approximately 63.4 degrees.
Angle of elevation from the boat to the top of the tree:
Let's draw a diagram of the situation:
C (person on boat)
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35 | \
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|__________\
A (top of tree)
In this diagram, C represents the person on the boat, A represents the top of the tree, and the lines connecting them form a right triangle. We want to find the angle of elevation, which is the angle between the horizontal line (from the person on the boat to the shore) and the line of sight from the person on the boat to the top of the tree. This is also angle θ in the diagram.
We know that the opposite side of this right triangle is 15 feet (the height of the tree) and the adjacent side is 35 feet (the horizontal distance from the person on the boat to the tree). Therefore:
- tan(θ) = opposite/adjacent = 15/35
Taking the arctangent of both sides gives us:
- θ = arctan(15/35) ≈ 23.1 degrees
Therefore, the angle of elevation from the boat to the top of the tree is approximately 23.1 degrees.