To find the angle formed between the top of the volleyball net pole and the rope, we can use the right triangle formed by the pole, the rope, and the horizontal distance from the base of the pole to the point where the rope is tied.
We can apply the inverse tangent (arctan) function, which relates the angles in a right triangle to the lengths of the opposite side and the adjacent side. In this case:
- The height of the pole (opposite side) = 8 feet
- The horizontal distance from the base of the pole to where the rope is tied (adjacent side) = 6 feet
The formula for the angle \( \theta \) is given by: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
Plugging in the values: \[ \tan(\theta) = \frac{8}{6} \]
Next, we can simplify this: \[ \tan(\theta) = \frac{4}{3} \]
Now, we can find the angle \( \theta \) using the inverse tangent (also called arctan): \[ \theta = \tan^{-1}\left(\frac{4}{3}\right) \]
Calculating the angle using a calculator: \[ \theta \approx 53.13^\circ \]
This angle is the angle between the height of the pole and the rope.
To find the angle formed between the top of the pole and the rope, we need to subtract the calculated angle \( \theta \) from \( 90^\circ \): \[ 90^\circ - 53.13^\circ \approx 36.87^\circ \]
Thus, the angle formed between the top of the pole and the rope is: \[ \approx 36.87^\circ \]
This matches your stated answer.