To find the height of the stone face, we can use trigonometry and break the problem down into two parts: finding the total height of the mountain up to the top of the face and the height up to the bottom of the face, then subtracting the two.

We are given:

• The distance from the observation point to the base of the mountain is $850$ feet.
• The angle of elevation to the bottom of the stone face is $28^\circ$.
• The angle of elevation to the top of the stone face is $31^\circ$.

Step 1: Find the height to the top of the stone face.

We can use the tangent function: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ Where:

• $\theta$ is the angle of elevation.
• The opposite side is the height (which we need to find).
• The adjacent side is the distance from the base of the mountain ($850$ feet).

For the top of the stone face, the angle is $31^\circ$, so: $\tan(31^\circ) = \frac{h_{\text{top}}}{850}$ Solving for $h_{\text{top}}$: $h_{\text{top}} = 850 \times \tan(31^\circ)$

Step 2: Find the height to the bottom of the stone face.

Similarly, for the bottom of the stone face, the angle is $28^\circ$, so: $\tan(28^\circ) = \frac{h_{\text{bottom}}}{850}$ Solving for $h_{\text{bottom}}$: $h_{\text{bottom}} = 850 \times \tan(28^\circ)$

Step 3: Find the height of the stone face.

The height of the stone face is the difference between the height to the top and the height to the bottom: $h_{\text{face}} = h_{\text{top}} - h_{\text{bottom}}$

Let's calculate this now.The height of the stone face is approximately $58.78$ feet.

Would you like further details or have any questions?

Here are 5 related questions:

1. How would the height change if the angles of elevation were increased?
2. How do trigonometric functions apply in real-world surveying problems?
3. How would the solution change if the observer's distance from the mountain was altered?
4. What happens if the mountain has a slope and is not perfectly vertical?
5. Can we find the horizontal distance if the stone face is observed from a different angle?

Tip: When using trigonometric functions in surveying problems, always ensure angles are in the correct units (degrees or radians) for calculations.