A communication tower stands on the edge of a cliff. At a horizontal distance of 150 m from the tower, the angle of elevation of the top of the tower is 65°30' and that of the foot of the tower is 45°15'. Find the height of the tower to the nearest tenth of a meter. Draw the triangle

### Problem Breakdown:

We are given the following:

1. **Horizontal distance from the observer to the tower**: 150 m.
2. **Angle of elevation to the top of the tower**: 65°30' (or 65.5° in decimal form).
3. **Angle of elevation to the foot of the tower**: 45°15' (or 45.25° in decimal form).

The goal is to find the height of the tower.

We will use trigonometry to solve the problem by splitting the height of the tower into two parts:

- The height from the foot of the tower to the observer’s eye level.
- The remaining height from the observer’s eye level to the top of the tower.

### Steps:

1. **Convert the angles**: 
   - Convert 65°30' to decimal: \( 65° + \frac{30}{60}° = 65.5° \).
   - Convert 45°15' to decimal: \( 45° + \frac{15}{60}° = 45.25° \).

2. **Using trigonometric relations (tangent function)**:
   - Let \( h_1 \) be the height from the observer’s eye level to the top of the tower.
   - Let \( h_2 \) be the height from the observer’s eye level to the foot of the tower.
   - The total height \( H = h_1 - h_2 \).

   Using the tangent function:
   \[
   \tan(65.5°) = \frac{h_1}{150}
   \]
   \[
   \tan(45.25°) = \frac{h_2}{150}
   \]

3. **Solve for \( h_1 \) and \( h_2 \)**:
   \[
   h_1 = 150 \times \tan(65.5°)
   \]
   \[
   h_2 = 150 \times \tan(45.25°)
   \]

4. **Calculate the total height**:
   \[
   H = h_1 - h_2
   \]

I’ll now calculate the values and provide the total height.

The total height of the communication tower is approximately **177.8 meters** to the nearest tenth of a meter.

Now for the triangle representation:
1. The observer's position is 150 meters horizontally away from the tower.
2. The angle of elevation to the foot of the tower is 45.25°, forming a right triangle where the vertical side is the height to the observer's eye level.
3. The angle of elevation to the top of the tower is 65.5°, with the full height of the tower as the vertical side of another right triangle.

Would you like to see the triangle drawing or need any further details?

### 5 Related Questions:
1. How does changing the angle of elevation affect the perceived height of the tower?
2. What would the total height be if the horizontal distance were halved?
3. How can we calculate the height of the cliff if the observer was at a higher altitude?
4. What are the applications of trigonometry in surveying communication towers?
5. How would we solve this if the angle of depression to the foot of the tower was given instead?

### Tip:
When dealing with angles of elevation or depression in real-world problems, always ensure your angles are measured correctly and converted to decimal form if needed for accuracy.

A communication tower stands on the edge of a cliff. At a horizontal distance of 150 m from the tower, the angle of elevation of the top of the tower is 65°30' and that of the foot of the tower is 45°15'. Find the height of the tower to the nearest tenth of a meter. Draw the triangle.

Find the height of a communication tower using trigonometry. With angles of elevation of 65.5° and 45.25° from a horizontal distance of 150 meters, this problem demonstrates how to calculate the total height using the tangent function. Ideal for students in Grades 10-12.

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