To find the height of the stone face carved on the side of the mountain, we can use trigonometry. Let’s break the problem down step by step.

Given Information:

• Distance from the base of the mountain: 650 feet
• Angle of elevation to the bottom of the face: 36°
• Angle of elevation to the top of the face: 39°

Step 1: Find the height to the bottom of the face

Using the tangent function:

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

$h_1 = 650 \tan(36^\circ)$

$h_1 = 650 \times 0.7265$

$h_1 \approx 472.2 \text{ feet}$

Step 2: Find the height to the top of the face

$h_2 = 650 \tan(39^\circ)$

$h_2 = 650 \times 0.8098$

$h_2 \approx 526.4 \text{ feet}$

Step 3: Calculate the height of the stone face

$\text{Height of the stone face} = h_2 - h_1$

$= 526.4 - 472.2$

$= 54.2 \text{ feet}$

Final Answer:

The height of the stone face is 54.2 feet.

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Would you like a more detailed breakdown or have any questions?

Here are some related questions:

1. How would the height change if the distance were different?
2. What if the angles of elevation were swapped?
3. How can this method be used to measure buildings?
4. Can we use sine or cosine instead of tangent in this case?
5. How accurate is this method in real-world applications?

Tip: Always ensure your calculator is in degree mode when working with angles in degrees!