To find the angle between the height of the tree (the vertical distance) and the owl's line of sight looking down at you, we can use the properties of a right triangle.
Let's define:
- \( h \): the height of the owl in the tree (which we need to find).
- \( d = 5 \) feet: the horizontal distance from you to the base of the tree.
- \( l = 14 \) feet: the diagonal line of sight from the owl to you.
In this scenario, the triangle formed has:
- The height \( h \) as the vertical side.
- The distance \( d \) (5 feet) as the horizontal side.
- The diagonal \( l \) (14 feet) as the hypotenuse.
We want to find the angle \( \theta \) between the height of the tree and the line of sight of the owl, which can be found using the inverse sine function:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{l} \]
However, first, we need to find the height \( h \) using the Pythagorean theorem:
\[ l^2 = h^2 + d^2 \]
Substituting the values:
\[ 14^2 = h^2 + 5^2 \]
Calculating the squares:
\[ 196 = h^2 + 25 \]
Now, solve for \( h^2 \):
\[ h^2 = 196 - 25 = 171 \]
Take the square root of both sides to find \( h \):
\[ h = \sqrt{171} \approx 13.08 \text{ feet} \]
Now, we can use this height to find the angle \( \theta \):
\[ \sin(\theta) = \frac{h}{l} = \frac{13.08}{14} \]
Calculating this value:
\[ \sin(\theta) \approx 0.9343 \]
Now, we use the inverse sine function to find the angle \( \theta \):
\[ \theta = \arcsin(0.9343) \]
Using a calculator to find the angle:
\[ \theta \approx 68.9^\circ \]
Thus, the angle between the height of the tree and the owl's line of sight looking down at you is approximately:
\[ \boxed{68.9^\circ} \]
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