Find the height of the tower using the information given in the illustration

To determine the height of the tower using the information given in the illustration:

1. **Given Information:**
   - Angle of elevation: \(85.952^\circ\)
   - Horizontal distance (base) from the observation point to the base of the tower: 80 feet.

2. **Relationship in the Triangle:**
   - The height of the tower corresponds to the side opposite the given angle.
   - The distance of 80 feet corresponds to the adjacent side of the right triangle.

3. **Using the Tangent Function:**
   \[
   \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
   \]
   \[
   \text{Height of the tower} = \text{Adjacent side} \times \tan(\theta)
   \]

   Substituting the known values:
   \[
   \text{Height} = 80 \times \tan(85.952^\circ)
   \]

As calculated earlier, the height is approximately **1130.44 feet**.

This is the height of the tower based on the given angle of elevation and the distance.

Would you like more details or have any other questions? Here are some related questions:

1. What other methods can be used to determine the height of a tower without using trigonometry?
2. How does the accuracy of the angle measurement affect the height calculation?
3. Can this method be used for determining the height of a tower from a different point of observation?
4. How would the calculation change if the angle were measured in radians instead of degrees?
5. What assumptions are we making in this calculation regarding the observer's position?

**Tip:** Always double-check the angle and distance measurements for accuracy when calculating heights in real-world scenarios.

Learn how to calculate the height of a tower using the angle of elevation (85.952°) and a horizontal distance of 80 feet. Follow a step-by-step solution using trigonometric principles. Suitable for Grades 10-12.

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