The top of a vertical pole 43 ft. high is sighted horizontally from a point 150 ft. from its base up the sloping site of a hill on which it rests. What angle does the sloping site make with the pole?

To solve this problem, we need to determine the angle between the sloping site and the vertical pole. We are given:

- The height of the pole is \( 43 \, \text{ft} \).
- The horizontal distance from the observer to the base of the pole is \( 150 \, \text{ft} \).

We will use the tangent function to find the angle \( \theta \) between the slope and the pole. Here's the reasoning:

### Step 1: Right Triangle Setup
The pole is vertical, and the observer sees it from a point that is horizontally \( 150 \, \text{ft} \) away from its base. The line from the observer to the top of the pole forms a right triangle. The vertical leg is the height of the pole \( 43 \, \text{ft} \), and the horizontal leg is \( 150 \, \text{ft} \). 

In this right triangle:
- The opposite side to the angle \( \theta \) is the height of the pole, which is \( 43 \, \text{ft} \).
- The adjacent side is the horizontal distance from the observer to the pole's base, which is \( 150 \, \text{ft} \).

### Step 2: Tangent Function
The tangent of the angle \( \theta \) is given by:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{43}{150}
\]

\[
\tan(\theta) = 0.2867
\]

### Step 3: Solving for \( \theta \)
To find the angle \( \theta \), take the arctangent (inverse tangent) of 0.2867:

\[
\theta = \tan^{-1}(0.2867)
\]

Using a calculator, we find:

\[
\theta \approx 16.04^\circ
\]

Thus, the angle that the sloping site makes with the pole is approximately \( 16.04^\circ \).

### Do you need more details or have any further questions?

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Here are 5 related questions to expand on this:

1. How would the angle change if the height of the pole increased?
2. What happens to the angle if the distance from the observer to the pole’s base is halved?
3. Can you calculate the hypotenuse of the triangle formed by the pole, base, and observer?
4. How would you adjust the calculation if the site had a horizontal angle different from 90°?
5. How does this concept relate to real-life applications in construction or surveying?

**Tip:** When dealing with trigonometric problems, always ensure you have a clear understanding of which sides of the triangle correspond to opposite, adjacent, and hypotenuse based on the angle you are trying to find.

Learn how to calculate the angle between a vertical pole and a slope using trigonometric principles. In this example, a 43 ft. tall pole is viewed from a distance of 150 ft. Discover how to apply the tangent function and arctangent to solve for the angle. Suitable for Grades 9-11.

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