To find the angle of the sail formed at the top of the vertical pole, we can utilize the tangent function from trigonometry. In the given scenario, we have a right triangle formed by the vertical pole (22 feet), the horizontal pole (8 feet), and the sail acting as the hypotenuse.
We can denote:
- The height of the vertical pole (opposite side) as \(a = 22\) feet.
- The length of the horizontal pole (adjacent side) as \(b = 8\) feet.
We need to find the angle \(\theta\) at the top of the vertical pole. This angle is adjacent to the horizontal pole. Using the tangent function:
\[ \tan(\theta) = \frac{a}{b} = \frac{22}{8} \]
Now we can calculate the angle \(\theta\):
\[ \tan(\theta) = \frac{22}{8} = 2.75 \]
To find \(\theta\), we take the arctangent (inverse tangent) of 2.75:
\[ \theta = \arctan(2.75) \]
Calculating \(\arctan(2.75)\):
Using a calculator:
\[ \theta \approx 70.68^\circ \]
Rounding to the nearest whole degree, the angle \(\theta\) is approximately:
\[ \theta \approx 71^\circ \]
Thus, the angle of the sail formed at the top of the vertical pole is approximately 71 degrees.