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, $156.08 \, \text{ft}$.

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