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A person is standing 300m away from a building with a flagpole on top. The angle from the person to the top of the building is 30°. The angle from the person to the top of the flagpole is 32°. Retrieve the height of the flagpole.

To retrieve the height of the flagpole, we can break the problem into two parts:

1. **Height of the building** using the angle of elevation to the top of the building.
2. **Height of the flagpole** using the angle of elevation to the top of the flagpole, and then subtracting the height of the building.

Here’s the step-by-step approach:

- **Known dataTo retrieve the height of the flagpole, we can break the problem into two parts:

1. **Height of the building** using the angle of elevation to the top of the building.
2. **Height of the flagpole** using the angle of elevation to the top of the flagpole, and then subtracting the height of the building.

Here’s the step-by-step approach:

- **Known **
  - Distance from the person to the base of the building: \( d = 300 \, \text{m} \)
  - Angle of elevation to the top of the building: \( \theta_1 = 30^\circ \)
  - Angle of elevation to the top of the flagpole: \( \theta_2 = 32^\circ \)

### Step 1: Find the height of the building

We can use the tangent of the angle to find the height of the building:

\[
\text{height of building (H)} = d \cdot \tan(\theta_1)
\]

Substituting the values:

\[
H = 300 \cdot \tan(30^\circ)
\]

\[
H = 300 \cdot 0.5774 = 173.22 \, \text{m}
\]

### Step 2: Find the total height (building + flagpole)

Now, use the angle of elevation to the top of the flagpole to find the total height of the building and flagpole together:

\[
\text{total height} = d \cdot \tan(\theta_2)
\]

Substituting the values:

\[
\text{total height} = 300 \cdot \tan(32^\circ)
\]

\[
\text{total height} = 300 \cdot 0.6249 = 187.47 \, \text{m}
\]

### Step 3: Find the height of the flagpole

The height of the flagpole is the difference between the total height and the height of the building:

\[
\text{height of flagpole} = 187.47 \, \text{m} - 173.22 \, \text{m}
\]

\[
\text{height of flagpole} = 14.25 \, \text{m}
\]

### Conclusion:
The height of the flagpole is approximately **14.25 meters**.

Would you like more details or have any questions?

### Follow-up questions:
1. What if the distance to the building changed? How would it affect the height calculations?
2. What is the effect of increasing the angle of elevation on the calculated height?
3. How would you calculate the flagpole height if the angle to the top of the building were unknown?
4. Can you use a different trigonometric function to find the same results? Why or why not?
5. How would a real-world measurement take into account inaccuracies or other factors like ground tilt?

### Tip:
When working with trigonometric functions, ensure your calculator is set to the correct mode (degrees or radians) based on the given angle information.

Learn how to calculate the height of a flagpole on top of a building using trigonometric principles. Given a 300-meter distance from the observer, a 30° angle to the top of the building, and a 32° angle to the top of the flagpole, we find the height using tangent functions. Suitable for students in Grades 9-12.

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