6. A flagpole broken over by the wind forms a right triangle with the ground. If the angle which the broken part makes with the ground is 50 deg and the distance from the tip of the pole to the foot is 55 feet, how tall was the pole?

This problem involves trigonometry, where a right triangle is formed by the flagpole, its broken section, and the ground. Here's how to solve it step-by-step:

### Problem Setup:
- Let the total height of the flagpole be \( H \).
- The flagpole is broken into two parts: 
  - The part that is still standing vertically, let's call this height \( h \).
  - The part lying on the ground forms an angle of 50° with the ground and touches the ground 55 feet away from the base of the pole. This distance is the horizontal leg of the triangle, which we will denote as \( x = 55 \, \text{feet} \).
- We are looking to find the total height \( H \) of the flagpole.

### Using Trigonometry:
We can apply the tangent function here, because tangent relates the angle of a right triangle to the opposite side (the height of the broken part of the flagpole) and the adjacent side (the horizontal distance):

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

In this case:
- \( \theta = 50^\circ \)
- The opposite side is the length of the broken part of the pole, which we’ll call \( b \).
- The adjacent side is the horizontal distance \( x = 55 \, \text{feet} \).

\[
\tan(50^\circ) = \frac{b}{55}
\]

### Solving for \( b \):
\[
b = 55 \times \tan(50^\circ)
\]

Let's compute that.The length of the broken part of the flagpole, \( b \), is approximately 65.55 feet.

### Total Height of the Pole:
Since the flagpole was originally standing vertically before it broke, the total height \( H \) is the sum of the still-standing part \( h \) and the broken part \( b \).

Since the flagpole broke at the ground level, the still-standing part is essentially 0. Therefore, the total height of the flagpole is approximately:

\[
H = b = 65.55 \, \text{feet}
\]

Thus, the original height of the flagpole was about 65.55 feet.

Would you like further details or have any questions?

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Here are 5 related questions to expand your understanding:
1. How would the problem change if the angle were increased to 60 degrees?
2. What would be the height if the horizontal distance was 100 feet?
3. Can you calculate the height using sine or cosine instead of tangent?
4. How does the tangent function behave as the angle approaches 90 degrees?
5. How can this problem be solved geometrically without trigonometric functions?

**Tip:** Always make sure to check the units of your answers, especially when dealing with feet, meters, or other distance units!

A flagpole broken over by the wind forms a right triangle with the ground. If the angle which the broken part makes with the ground is 50° and the distance from the tip of the pole to the foot is 55 feet, how tall was the pole?

Discover how to calculate the height of a broken flagpole using trigonometry. Given a 50° angle between the broken part and the ground and a 55-foot horizontal distance, this solution explains the tangent function step by step. Suitable for high school students (Grades 9-12).

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