Rico secures a volleyball-net pole to the ground with a rope that is attached to the top of the pole and creates a diagonal distance to the ground. The volleyball-net pole creates a 90 ° angle to the ground. If the pole is 8 feet in height and Rico ties the rope five feet from the base of the pole, what is the angle formed between the top of the pole and the rope? Round your answer to the nearest whole degree. (1 point)

Question

Responses

33 degrees
33 degrees

32 degrees
32 degrees

31 degrees
31 degrees

34 degrees
34 degrees

Answer 1

To find the angle formed between the top of the pole and the rope, we can use trigonometry.

Let's define the scenario with some triangles:

1. The height of the pole is 8 feet, which is the vertical side (opposite).
2. The distance from the base of the pole to the point where the rope is tied is 5 feet, which is the horizontal side (adjacent).
3. The rope acts as the hypotenuse of the right triangle formed.

We can use the tangent function, which relates the opposite side to the adjacent side in a right triangle:

\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{8}{5} \]

Now, let's calculate the angle \( \theta \):

\[ \theta = \tan^{-1}\left(\frac{8}{5}\right) \]

Calculating this gives:

\[ \theta \approx \tan^{-1}(1.6) \approx 57.99^{\circ} \]

However, we want the angle formed between the top of the pole and the rope, which is complementary to angle \( \theta \):

\[ \text{Angle needed} = 90^{\circ} - \theta \]

Calculating this:

\[ \text{Angle needed} = 90^{\circ} - 57.99^{\circ} \approx 32.01^{\circ} \]

Rounding to the nearest whole degree:

\[ \text{Angle needed} \approx 32^{\circ} \]

Thus, the angle formed between the top of the pole and the rope is 32 degrees.