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The problem you provided involves finding the height of a post based on two different angles of elevation and distances. Let's solve it step by step.

### Problem Recap:

1. You observe a post with an angle of elevation \( \alpha \) such that \( \tan(\alpha) = \frac{1}{6} \).
2. You then move 30 meters closer to the post, and the angle of elevation becomes \( 45^\circ \).
3. The task is to find the height of the post.

### Step-by-Step Solution:

#### Step 1: Use the tangent of the angle \( \alpha \).
Given:
\[
\tan(\alpha) = \frac{1}{6}
\]
Let the initial distance from the post be \( d \) meters, and the height of the post be \( h \) meters. Then,
\[
\tan(\alpha) = \frac{h}{d} = \frac{1}{6}
\]
Thus,
\[
h = \frac{d}{6}
\]

#### Step 2: Use the tangent of the second angle \( 45^\circ \).
When the observer moves 30 meters closer, the new distance from the post becomes \( d - 30 \) meters.

Given that \( \tan(45^\circ) = 1 \), we have:
\[
\tan(45^\circ) = \frac{h}{d - 30} = 1
\]
So,
\[
h = d - 30
\]

#### Step 3: Equate the two expressions for \( h \).
From the two equations:
1. \( h = \frac{d}{6} \)
2. \( h = d - 30 \)

We equate them:
\[
\frac{d}{6} = d - 30
\]

#### Step 4: Solve for \( d \).
To solve for \( d \), multiply the entire equation by 6 to eliminate the fraction:
\[
d = 6(d - 30)
\]
Expanding and simplifying:
\[
d = 6d - 180
\]
\[
180 = 5d
\]
\[
d = 36 \text{ meters}
\]

#### Step 5: Calculate the height \( h \) of the post.
Now that we know \( d = 36 \) meters, substitute it back into the equation for \( h \):
\[
h = \frac{d}{6} = \frac{36}{6} = 6 \text{ meters}
\]

### Final Answer:
The height of the post is **6 meters**.

Would you like any additional details, or do you have any further questions?

### Related Questions:
1. How would the problem change if the angle of elevation were given as \( \tan(\alpha) = \frac{1}{4} \)?
2. What if the observer moved closer by a different distance, say 50 meters instead of 30 meters?
3. Can you generalize a formula to find the height of any post given any angles and distances?
4. What if the post were on an incline? How would that affect the angle calculations?
5. How does changing the initial observation distance affect the height calculation?
6. What would the height be if the angle \( \alpha \) were changed to \( 30^\circ \) instead?
7. How could you solve this problem graphically?
8. How does the problem illustrate the use of trigonometric identities?

### Tip:
Always ensure to use the correct trigonometric identities when working with angles and distances, especially in word problems like this one!

Learn how to find the height of a post using trigonometry with a detailed step-by-step solution. This problem involves angles of elevation, distances, and the tangent function. Suitable for high school students.

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