Height of Mt. Everest The highest mountain peak in the world...

Height of Mt. Everest 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 . This difficult measurement had to be done from a great distance. At an altitude of $14,545 \mathrm{ft}$ on a different mountain, the straight-line distance to the peak of Mt. Everest is $27.0134 \mathrm{mi}$ and its angle of elevation is $\theta=5.82^{\circ} .$ (Source: Dunham, W., The Mathematical Universe, John Wiley and Sons.) (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?

Height of Mt. Everest 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 . This difficult measurement had to be done from a great distance. At an altitude of
$14,545 \mathrm{ft}$ on a different mountain, the straight-line distance to the peak of Mt. Everest
is $27.0134 \mathrm{mi}$ and its angle of elevation is $\theta=5.82^{\circ} .$ (Source: Dunham, W., The Mathematical Universe, John Wiley and Sons.)
(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

Effect of Earth’s Curvature on Measurements

When measurements are taken over large distances, the curvature of the Earth needs to be considered because it can alter the apparent heights and distances due to the Earth's spherical shape. Neglecting the Earth's curvature in such cases can lead to errors, as it affects the line of sight and the effective distance between the observer and the object being surveyed.

Trigonometry in Surveying

This concept involves using trigonometric functions to determine unknown distances or heights by measuring angles and a baseline distance. It is a fundamental technique in surveying, where observers take measurements from a known point and use relationships in right triangles to calculate the dimensions of inaccessible objects or distances on large scales.

Angle of Elevation

The angle of elevation is the angle between the horizontal line of sight and the line of sight up to an object. In surveying, measuring the angle of elevation allows for the creation of a right triangle where the opposite side represents the vertical difference, enabling one to use trigonometric ratios to determine heights or distances.

Tangent Function in Right Triangles

In the context of right triangles, the tangent function relates the angle of elevation to the ratio of the opposite side (height difference) over the adjacent side (horizontal distance). This is particularly useful in practical applications like calculating the height of a mountain when the distance and angle are known.

Transcript

00:01 This problem explores trigonometric functions and solving different kind of situations using right angle applications.

00:10 In this specific case, as you can see in my, or you probably can't see in my terrible drawing, but we're going to try to solve this problem here.

00:20 You're on some mountain here, which is very far away from mount everest, right? so this is mount everest here.

00:32 And we're on this location, which has some height.

00:35 Let's say this is going to be a height of, let's say, 10 ,000.

00:42 So this is 10 ,000 feet here.

00:46 And we know what this distance is here to the peak of mount everest.

00:52 So let's say that's 30 miles.

00:55 I'm just making these up as i go, but you should be able to extrapolate this out to a different kind of problem.

01:03 So once we're on this mountain, and we have.

01:07 The distance to the peak of this mountain and we know this angle here is going to be theta is let's say 10 degrees now we want to find the so we're given the hypotenuse right so we're given the long side so we want to find this vertical side here and we have this angle here and we can do that using this handy dandy tool which carries you through all of trigonometry called sokatoa so we're given this angle right so everything that we look at is going to be with reference to this angle.

01:42 So sokatoa, if you haven't heard of it, is sine, cosine, and tangent are the s, c, and t.

01:48 And the o here is opposite.

01:51 H is hypotenuse.

01:53 A is adjacent.

01:54 So this is our a, right? so this angle here.

01:59 So this is the adjacent angle, or the next two angle is how i remember it.

02:04 The opposite, right, opposite the angle, angles on this side, so this side is opposite, that's an o, and this side is obviously the hypotenus here.

02:15 And we're given this angle here.

02:16 So we want to find this.

02:18 So the angles, the sides that we're concerned with is in o and an h.

02:23 So which ones relate to o and h? that's going to be sign.

02:26 So we can actually just do a sign of our theta, which is 10.

02:36 And that is equal to again, it's o over h or a over h...

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