To solve this problem, we can use trigonometry, specifically the tangent function, which is given by:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Where:

• $\theta$ is the angle of elevation,
• "opposite" is the height of the point you're measuring to,
• "adjacent" is the horizontal distance from the point of observation.

Steps to find the height of the stone face:

1. Calculate the height to the top of the stone face:

   • The angle of elevation to the top is 43°.
   • The distance from the base of the mountain is 650 feet.
   • Using the tangent function:

   $\tan(43^\circ) = \frac{h_{\text{top}}}{650}$

   Solving for $h_{\text{top}}$:

   $h_{\text{top}} = 650 \times \tan(43^\circ)$

2. Calculate the height to the bottom of the stone face:

   • The angle of elevation to the bottom is 40°.
   • Again, using the tangent function:

   $\tan(40^\circ) = \frac{h_{\text{bottom}}}{650}$

   Solving for $h_{\text{bottom}}$:

   $h_{\text{bottom}} = 650 \times \tan(40^\circ)$

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:

   $\text{Height of the stone face} = h_{\text{top}} - h_{\text{bottom}}$

Now, let's calculate the values:

1. $h_{\text{top}} = 650 \times \tan(43^\circ)$
2. $h_{\text{bottom}} = 650 \times \tan(40^\circ)$

Let me compute the values. math

Given values

distance = 650 # distance from the base in feet angle_top = 43 # angle of elevation to the top in degrees angle_bottom = 40 # angle of elevation to the bottom in degrees

Convert degrees to radians for calculation

angle_top_rad = math.radians(angle_top) angle_bottom_rad = math.radians(angle_bottom)

Calculate heights using tangent function

height_top = distance * math.tan(angle_top_rad) height_bottom = distance * math.tan(angle_bottom_rad)

Calculate the height of the stone face

stone_face_height = height_top - height_bottom height_top, height_bottom, stone_face_heightThe height of the stone face is approximately 60.72 feet.

Here's the breakdown:

• The height to the top of the stone face is about 606.13 feet.
• The height to the bottom of the stone face is about 545.41 feet.
• The difference between these gives the height of the stone face, which is approximately 60.72 feet.

Would you like further details or have any other questions?

Related questions:

1. How does the angle of elevation affect the height calculation?
2. What would happen if the distance from the base was different?
3. How could we apply the same method to calculate the height of a building?
4. What other trigonometric functions could be used in this kind of problem?
5. How can we use a calculator to find the tangent of an angle?

Tip: Always ensure to convert angles from degrees to radians when using trigonometric functions in calculators that require radians.