A water tower 30 m tall is located at the top of a hill. From a distance of D = 115 m down the hill it is observed that the angle formed between the top and base of the tower is 8°. Find the angle of inclination of the hill. (Round your answer to one decimal place.)

To find the angle of inclination of the hill, we can use trigonometry. We have a right triangle formed by the height of the water tower, the distance from the observer to the base of the tower, and the line of sight to the top of the tower.

1. **Height of the tower (h)**: 30 m
2. **Distance from the observer to the base of the tower (D)**: 115 m
3. **Angle of elevation (θ)**: 8°

### Step 1: Find the horizontal distance from the observer to the base of the hill (x).

Using the tangent of the angle of elevation:

\[
\tan(θ) = \frac{h}{x}
\]

\[
x = \frac{h}{\tan(θ)} = \frac{30}{\tan(8°)}
\]

Calculating \(x\):

\[
x ≈ \frac{30}{0.1405} ≈ 213.5 \text{ m}
\]

### Step 2: Find the total distance down the hill (D + x).

\[
\text{Total distance down the hill} = D + x = 115 + 213.5 ≈ 328.5 \text{ m}
\]

### Step 3: Use the sine function to find the angle of inclination (α) of the hill.

In this triangle, we have:

\[
\sin(α) = \frac{h}{\text{Total distance down the hill}} = \frac{30}{328.5}
\]

Calculating \(α\):

\[
α = \arcsin\left(\frac{30}{328.5}\right)
\]

Calculating this gives:

\[
α ≈ \arcsin(0.0912) ≈ 5.2°
\]

### Final Answer

The angle of inclination of the hill is approximately **5.2°** (rounded to one decimal place).

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

1. How would the angle change if the height of the tower were different?
2. What would happen to the angle if the distance down the hill increased?
3. How can this setup be applied to other real-world scenarios?
4. What other trigonometric functions could be used in this problem?
5. How would you solve this problem using different units (e.g., feet)? 

**Tip**: Always draw a diagram when dealing with trigonometric problems to visualize the relationships better.

Discover how to calculate the angle of inclination of a hill based on a water tower's height and the angle of elevation observed from a distance. This problem utilizes trigonometric concepts, with a tower height of 30 m and an observation distance of 115 m, resulting in an inclination angle of approximately 5.2°.

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