Question

Rico secures a vollyball-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 hight and Rico ties the rope 6 feet from the base of the pole, what is the angle formed between the top of the pole and the rope?  Show all your work in order to receive full credit for the problem, including your original equation and your answer rounded to the nearest whole number.

Answer 1

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

Step 1: Identify the components.

• Height of the pole (opposite side): \( h = 8 \) feet
• Horizontal distance from the base of the pole to where the rope is tied (adjacent side): \( d = 6 \) feet

Step 2: Use the tangent function.

The tangent of the angle \( \theta \) we want to find is given by the ratio of the height of the pole (opposite) to the horizontal distance (adjacent):

\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d} = \frac{8}{6} \]

Step 3: Calculate \( \tan(\theta) \).

\[ \tan(\theta) = \frac{8}{6} = \frac{4}{3} \approx 1.3333 \]

Step 4: Use the arctangent function to find the angle.

To find the angle \( \theta \):

\[ \theta = \tan^{-1}\left(\frac{4}{3}\right) \]

Using a calculator:

\[ \theta \approx 53.13^\circ \]

Step 5: Round the angle to the nearest whole number.

Rounding \( 53.13^\circ \) gives us:

\[ \theta \approx 53^\circ \]

Final Answer:

The angle formed between the top of the pole and the rope is approximately \( 53^\circ \).