Problem 58 - A Graphical Approach to Precalculus with Limits

Question

The highest mountain peak in the world is Mt. Everest, located in the Himalayas. The height of this enormous mountain was determined in 1856 by surveyors using trigonometry long before the peak was first climbed in \(1953 .\) At an altitude of 14.545 feet on a different mountain, the straight-line distance to the peak of Mt. Everest is 27.0134 miles and its angle of elevation is \(\theta=5.82^{\circ} .\) (a) Approximate the height (in feet) of Mt. Everest. (b) In the actual measurement, Mt. Everest was over 100 miles away and the curvature of Earth had to be taken into account. Would the curvature of Earth make the peak appear taller or shorter than it actually is?

Step-by-Step Solution

Answer

(a) The height of Mt. Everest is approximately 29,102 feet. (b) The curvature of Earth makes the peak appear shorter.

Step 1: Convert the Distance from Miles to Feet

We know that 1 mile is equal to 5280 feet. Therefore, to convert 27.0134 miles to feet, we use the conversion factor:\[ 27.0134 \text{ miles} \times 5280 \text{ feet/mile} = 142,629.072 \text{ feet} \]

Step 2: Use the Angle of Elevation to Find the Height Difference

The angle of elevation \(\theta = 5.82^\circ\) allows us to set up a right triangle where the opposite side is the height difference between the observation point and the peak of Everest. Using the tangent function:\[ \tan(\theta) = \frac{\text{height difference}}{\text{distance}} \]\[ \tan(5.82^\circ) = \frac{\text{height difference}}{142,629.072 \text{ feet}} \]Solving for the height difference gives:\[ \text{height difference} = \tan(5.82^\circ) \times 142,629.072 \text{ feet} \approx 14,557.28 \text{ feet} \]

Step 3: Add the Height of the Observation Point

The observation point is at an altitude of 14,545 feet. Adding this to the height difference calculated in the previous step will give us the total height of Mt. Everest:\[ \text{Total height} = 14,545 \text{ feet} + 14,557.28 \text{ feet} = 29,102.28 \text{ feet} \]

Step 4: Consider the Earth's Curvature Affect on Perceived Height

The curvature of the Earth affects measurements by making distant objects appear lower than they are. Without considering this effect, the calculated height might be less than the actual height, so the curvature would make the peak appear shorter than it actually is.

Key Concepts

Angle of Elevation

The angle of elevation is a key concept in trigonometry, helping us measure the height of distant objects like mountains. This is the angle formed between the horizontal line of sight and the line of sight up to the object. In our problem, the angle is given as \(\theta = 5.82^\circ\).
This means that when surveyors were positioned on another mountain, they used this angle to look up towards the summit of Mt. Everest.
This measurement serves as the basis for setting up a right triangle, essential for calculating the mountain's height using trigonometric functions like the tangent function.

• It helps in indirect measurement methods, crucial where direct measurements are not possible.
• It's essential in various real-world applications, from simple height calculations to advanced navigations.

Understanding the angle of elevation can greatly aid in solving problems involving height and distance in a straightforward manner.

Right Triangle

Right triangles play a foundational role in trigonometry and practical measurements. In this scenario, the triangle formed includes the line of sight, the "height difference" from the observer's position to the peak, and the horizontal distance converted from miles to feet.
The right angle in this triangle is between the horizontal ground distance and the imaginary vertical line drawn from the observer's point to the mountain peak.
This right triangle is pivotal because it allows us to apply trigonometric functions, ensuring calculations are precise.

• Right triangles have one angle exactly \(90^\circ\), simplifying calculations.
• They help relate multiple parts of a triangle using functions like sine, cosine, and tangent.
• The Pythagorean theorem applies, offering another tool for solving side lengths.

The principles of right triangles simplify complex problems, making it much easier to find unknown distances or heights.

Tangent Function

The tangent function serves as a powerful tool in trigonometry, especially useful when dealing with right triangles and elevation problems. In this particular problem, the tangent of the elevation angle was used to determine the height difference between the observation point and the peak of Mt. Everest.
The formula used is:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
Where "opposite" refers to the height difference and "adjacent" refers to the horizontal distance from the observation point to the mountain.
Inserting the values, the formula becomes:
\[ \tan(5.82^\circ) = \frac{\text{height difference}}{142,629.072 \text{ feet}} \]
Solving this gives us the height difference needed to further compute the total height of the mountain.

The tangent function's practicality comes from its ability to relate angles to the ratio of side lengths in a right triangle. It's a crucial component in fields ranging from architecture to astronomy, always valuable where measurements based on angles are required.

Curvature of the Earth

When measuring vast distances, like the height of Mt. Everest, the Earth's curvature can affect the accuracy of measurements. As Earth is round, objects lying far on the horizon, like Mt. Everest, can appear slightly lower than they naturally are.
In historical surveys where areas span significant distances, accounting for Earth's round shape is crucial.
In the solution provided, it was noted that without considering the curvature, the peak would seem to be shorter than its actual height.

• Curvature adjustments ensure more accurate distance and height calculations.
• They are vital in geodesy, the science of accurately measuring Earth's shape, orientation, and gravity field.
• Considering curvature is important even in designing structures like bridges and tunnels.

By incorporating the curve of Earth into measurements, surveyors ensure they obtain more precise results that reflect true physical dimensions, avoiding potential discrepancies in data collected over long distances.