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The problem you uploaded involves finding the height of a tower using the angles of elevation from two points on a straight road at a distance \( x \) from each other. The angles of elevation of the top of the tower from these two points are \( \alpha \) and \( \beta \), with \( \alpha > \beta \).

### Formula Derivation:

1. Let the height of the tower be \( h \).
2. Let the distance from the first point (where the angle of elevation is \( \alpha \)) to the base of the tower be \( d \), and the distance from the second point (where the angle of elevation is \( \beta \)) to the base of the tower be \( d + x \).

Using the tangent formula for the angles of elevation:
- From the first point:
  \[
  \tan(\alpha) = \frac{h}{d}
  \]
  Hence,
  \[
  d = \frac{h}{\tan(\alpha)}
  \]

- From the second point:
  \[
  \tan(\beta) = \frac{h}{d + x}
  \]
  Hence,
  \[
  d + x = \frac{h}{\tan(\beta)}
  \]
  So,
  \[
  d = \frac{h}{\tan(\beta)} - x
  \]

### Solving for \( h \):

Equating the two expressions for \( d \):
\[
\frac{h}{\tan(\alpha)} = \frac{h}{\tan(\beta)} - x
\]

Multiply both sides by \( \tan(\alpha) \tan(\beta) \):
\[
h (\tan(\beta) - \tan(\alpha)) = -x \tan(\alpha) \tan(\beta)
\]

Therefore, solving for \( h \):
\[
h = \frac{x \cdot \tan(\alpha) \cdot \tan(\beta)}{\tan(\alpha) - \tan(\beta)}
\]

### Correct Answer:

The height of the tower \( h \) is given by:
\[
h = \frac{x \cdot \tan(\alpha) \cdot \tan(\beta)}{\tan(\alpha) - \tan(\beta)}
\]

This corresponds to option **A** in the image.

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Do you want further details or have any questions about the problem?

Here are five related questions to extend your understanding:

1. How would the height of the tower change if the distance between the two points is halved?
2. Can you derive the formula for height if the two points are on opposite sides of the tower?
3. How can you apply this method to calculate the height of a building or tree in real life?
4. What happens if \( \alpha \) is less than \( \beta \)? Does the formula still hold?
5. How can we measure angles of elevation in practical scenarios?

**Tip:** Always remember to ensure that the angles of elevation are measured from the horizontal plane for correct calculations in height-related problems.

Two points on a straight road to the base of a tower are at a distance x from each other and on the same side of the tower. The angles of elevation of the top of the tower from the two points are α and β. What is the height of the tower? (α > β)

Learn how to calculate the height of a tower using trigonometry by understanding the angles of elevation from two different points. This problem involves key concepts like trigonometric ratios and the formula for height calculation, suitable for students in Grades 9-10.

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