Solve each problem involving an angle of elevation or...

Solve each problem involving an angle of elevation or depression. 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 it was first climbed in 1953 . (IMAGES CANNOT COPY) (a) Approximate the height (in feet) of Mt. Everest. (b) In the actual measurement, Mt. Everest was over $100 \mathrm{mi}$ 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?

Solve each problem involving an angle of elevation or depression.
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 it was first climbed in 1953 .
(IMAGES CANNOT COPY)
(a) Approximate the height (in feet) of Mt. Everest.
(b) In the actual measurement, Mt. Everest was over $100 \mathrm{mi}$ 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?

Key Concepts

Angle of Elevation

The angle of elevation is the angle formed between the horizontal line from the observer and the line of sight upward to an object. This concept is crucial when determining heights or distances using trigonometry, as the observer's line of sight forms one side of a right triangle with the height or distance as another side.

Trigonometric Functions in Right Triangles

Trigonometric functions, particularly the tangent function, are commonly used in right triangle calculations related to angles of elevation and depression. The tangent function relates the opposite side (such as the height of a mountain) to the adjacent side (such as the distance from the observer), allowing for the computation of unknown measurements when an angle and one side are known.

Trigonometry in Surveying

Trigonometry is a fundamental tool in surveying, where it is used to measure distances and heights that are difficult to reach directly. By using angles of elevation or depression and applying trigonometric ratios, surveyors can calculate the dimensions of objects or geographical features, like mountains, even when these features are far away or obscured from direct measurement.

Curvature of the Earth

The curvature of the Earth must be considered in long-distance measurements because it can affect the apparent height and distance of observed objects. When surveying from great distances, such as when measuring a mountain peak, the Earth's curvature may cause the actual line of sight to deviate from a straight horizontal line, potentially making the peak appear slightly shorter than its true height.

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