We can solve this problem using trigonometry, specifically the sine function in a right triangle.

Step 1: Define the given values

• The observer's altitude: $14,586$ ft
• The straight-line distance to Mountain A: $27.9331$ miles
• The angle of elevation: $\theta = 5.59^\circ$
• We need to find the height of Mountain A.

Step 2: Convert miles to feet

Since 1 mile = 5,280 feet, we convert the straight-line distance: $27.9331 \times 5280 = 147,505.37 \text{ feet}$

Step 3: Use the sine function

In the right triangle formed, the height difference ($h$) between Mountain A's peak and the observer is: $h = d \cdot \sin(\theta)$ $h = 147,505.37 \times \sin(5.59^\circ)$

Using a calculator:

$h = 147,505.37 \times 0.09742$ $h \approx 14,371 \text{ feet}$

Step 4: Find the total height of Mountain A

Since the observer is already at 14,586 feet, the total height of Mountain A is:

$\text{Total height} = 14,586 + 14,371$

$\approx 28,957 \text{ feet}$

Final Answer:

The height of Mountain A is 28,957 feet.

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Part (b): Effect of Earth's Curvature

When the observer is over 100 miles away, the curvature of Earth must be considered. The curvature of Earth would make the peak appear shorter than it actually is because the Earth's surface curves downward, making distant objects appear lower than they truly are.

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Final Answers:

(a) The height of Mountain A is 28,957 feet.
(b) The curvature of Earth would make the peak appear shorter than it actually is.

Would you like a more detailed explanation of any step?

Here are some related questions to expand on this topic:

1. How does the Earth's curvature affect long-distance altitude measurements?
2. How can we adjust for Earth's curvature in elevation measurements?
3. What trigonometric function would be used if we were given the horizontal distance instead?
4. What would the error be if we ignored Earth's curvature in this measurement?
5. Can atmospheric refraction impact elevation measurements as well?

Tip: When dealing with trigonometric calculations, always convert all units to a consistent system (e.g., feet or meters) before computing! 🚀