9.1.3. Example 3

An observer 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from her eyes is 45°.

In this scenario, we have an observer who is standing away from a chimney. The observer's height is 1.5 meters, and she is located 28.5 meters away from the base of the chimney. The angle created between her line of sight to the top of the chimney and the horizontal line from her eyes to the base of the chimney is called the angle of elevation. Here, this angle is 45 degrees.

Imagine you are standing in a park looking up at a tall slide. The distance you are standing from the slide is like the distance to the chimney, and the height of the slide represents the height of the chimney. If you look up at the top of the slide at a 45-degree angle, that’s similar to what the observer is experiencing.

Here, AB is the chimney, CD the observer, and ∠ADE the angle of elevation. In this case, ADE is a triangle, right-angled at E and we are required to find the height of the chimney.

We can visualize this scenario using a right triangle. In this case, AB represents the height of the chimney, and CD is the observer's line at eye level. The angle of elevation ∠ADE indicates the angle formed by the horizontal line of the observer's sight and the line extending from her eyes up to the top of the chimney. Since we know one angle (45 degrees) and the two sides opposite and adjacent to it (the observer's distance from the chimney and the height we need to find), we can use trigonometry to solve for AB.

Think of the scenario as trying to find how high a tree is from a specific distance. If you stand at a certain spot and stare upwards at the top of the tree, you create a triangle with the ground and your line of sight. Just like in our triangle situation, we can use objects readily available in the park to visualize angles and heights.

We have AB = AE + BE = AE + 1.5 and DE = CB = 28.5 m. To determine AE, we choose a trigonometric ratio, which involves both AE and DE. Let us choose the tangent of the angle of elevation.

In this part, we express the height of the chimney (AB) in terms of AE, which is the height from the observer's eye level to the top of the chimney. We note that BE is the observer's height (1.5 m), so AB equals AE plus 1.5 m. We then focus on triangle ADE and use the tangent of the angle of elevation to relate AE and DE. The tangent of an angle in a right triangle gives us the ratio of the opposite side to the adjacent side. Since the angle is 45 degrees, we use the tangent function: tan(45°) = AE/DE.

If you think back to the tree example, if you wanted to know how high the tree is without climbing it, you could measure a distance from the base and then calculate the height using the angle you see it from. In this case, the height corresponded with the vertical rise (height of the tree) and the horizontal distance you walked away from it.

Now, tan 45° = AE/DE, i.e., 1 = AE/28.5. Therefore, AE = 28.5.

Here, we calculate AE more explicitly using the fact that the tangent of 45 degrees equals 1. This means that the height AE is equal to the distance DE (28.5 m) because the tangent ratio gives us a simplistically direct relationship in this case. Thus, we conclude that AE is 28.5 m.

Going back to our tree analogy, finding AE is like realizing that for every meter you stand from the tree to see it, the height goes up by that same ratio because the angle creates an even slope.

So the height of the chimney (AB) = (28.5 + 1.5) m = 30 m.

Finally, we calculate the total height of the chimney (AB) by adding the height from the observer's eye level (AE) to the observer's height (1.5 m). Therefore, AB equals 28.5 meters plus 1.5 meters, resulting in a total chimney height of 30 meters.

This can be compared to measuring the total height of a tree after determining how high one could see from a point below. You measure from where your eyes are level and add your height to grasp the total height of the tree, which is how we concluded the overall height of the chimney.

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