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?

In this problem, we are tasked with finding the angle between the vertical pole and the sloping site of the hill.

Here are the key points:
- The pole is vertical and 43 feet high.
- The observer is horizontally 150 feet away from the base of the pole, but the observer is up a sloping hill.
- We need to find the angle that the sloping hill (site) makes with the vertical pole.

### Step-by-Step Solution:

This is a case of applying trigonometry, where we are trying to determine the angle between the sloping ground and the vertical pole.

1. **Identify the Components:**
   - The vertical height of the pole is \( 43 \, \text{ft} \).
   - The horizontal distance from the base of the pole to the observer is \( 150 \, \text{ft} \).
   - The hill is sloped, so we need to consider the actual distance from the observer's point on the hill to the base of the pole, which we'll call \( d \).

2. **Calculate the Hypotenuse (Distance \( d \)):**

Since the observer's view is horizontal from a point up the hill, the distance \( d \) from the observer to the base of the pole forms the hypotenuse of a right triangle, where:
- The opposite side is the height of the pole \( 43 \, \text{ft} \),
- The adjacent side is the horizontal distance \( 150 \, \text{ft} \).

We can calculate the hypotenuse using the Pythagorean theorem:

\[
d = \sqrt{(43)^2 + (150)^2}
\]

\[
d = \sqrt{1849 + 22500} = \sqrt{24349} \approx 156.08 \, \text{ft}
\]

So, the actual distance from the observer to the base of the pole is approximately \( 156.08 \, \text{ft} \).

3. **Determine the Angle \( \alpha \):**
The sloping site of the hill forms an angle with the vertical pole. We need to calculate the angle \( \alpha \) between the vertical pole and the slope.

For this, we can use the sine function because:
- The opposite side is the vertical height of the pole, \( 43 \, \text{ft} \),
- The hypotenuse is the distance we just calculated, \( 156.08 \, \text{ft} \).

\[
\

Learn how to find the angle between a vertical pole and a sloping hill using trigonometric concepts. This problem involves applying the Pythagorean theorem and the sine function to a right triangle with a pole height of 43 ft and a horizontal distance of 150 ft. Suitable for students in Grades 9-12.

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