Let's analyze the problem and solve it step by step.

Given:

• Height of the lighthouse, $h = 25$ feet.
• Distance from the foot of the lighthouse to a point on the hill, $d = 80$ feet.
• Angle subtended by the lighthouse at that point, $\theta = 17^\circ$.

We need to find the angle $\alpha$ that the side of the hill makes with the lighthouse.

Step 1: Understanding the Geometry

Consider a right triangle where:

• The height of the lighthouse $h = 25$ feet is the vertical side.
• The distance $d = 80$ feet is along the hill, forming the hypotenuse of the triangle.
• The angle $\theta = 17^\circ$ is the angle subtended by the lighthouse from that point.

Step 2: Finding the Angle of Elevation

Using the formula for the tangent in a right triangle: $\tan(\theta) = \frac{h}{x}$ Where:

• $\theta = 17^\circ$
• $h = 25$ feet
• $x$ is the horizontal distance from the foot of the lighthouse to the point on the hill.

So, $\tan(17^\circ) = \frac{25}{x}$

$x = \frac{25}{\tan(17^\circ)}$

Now, calculate $x$: $x \approx \frac{25}{0.3057} \approx 81.76 \text{ feet}$

Step 3: Finding the Angle $\alpha$

The side of the hill forms the hypotenuse $d$ in the triangle. To find the angle $\alpha$ between the hill and the horizontal plane, we use:

[ \cos(\alpha) = \frac