9.1.1. Example 1

A tower stands vertically on the ground. From a point on the ground, which is 15 m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 60°. Find the height of the tower.

In this problem, we have a vertical tower and a point on the ground 15 meters away from it. The angle of elevation means how high from the horizontal line the top of the tower appears to the observer at that point. Here, the angle of elevation to the top of the tower is 60 degrees, and we need to calculate the height of the tower, which we will refer to as 'AB'.

Think of standing at a distance from a tall building. When you look up at the top of the building, you can imagine drawing an angle from your eyes to the top of the building; this is similar to the angle of elevation in the problem.

First let us draw a simple diagram to represent the problem (see Fig. 9.4). Here AB represents the tower, CB is the distance of the point from the tower and angle ACB is the angle of elevation.

To visualize the problem, we can draw a right triangle. In this triangle, one side (AB) represents the height of the tower, while the other side (BC) represents the horizontal distance from the observer to the tower (15 m). The angle ACB is the angle of elevation (60°). The triangle formed helps us to apply trigonometry to find the height of the tower.

Imagine a ladder leaning against a wall. The height the ladder reaches on the wall is like the height of the tower, and the distance the base of the ladder is from the wall is like the 15 m in our problem.

Also, ACB is a triangle, right-angled at B. To solve the problem, we choose the trigonometric ratio tan 60° (or cot 60°), as the ratio involves AB and BC.

The triangle ACB is identified as a right triangle because one angle measures 90 degrees at point B. Here, we will use the tangent function, which relates the opposite side (height of the tower, AB) to the adjacent side (distance to the tower, BC). The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.

If you’ve ever used a ladder, think about how when you measure the height you reach on a wall (the height of the ladder against the wall, here represented by AB), you're finding a relationship between how high you can reach and how far the base of the ladder is from the wall (like BC).

Now, tan 60° = AB / BC i.e., 3 = AB / 15 i.e., AB = 15 * 3.

Using the tangent function for 60 degrees, we know tan 60° = √3 (or approximately 1.732). In this case, we set up the equation as tan 60° = AB / 15. By multiplying both sides by 15, we find that the height of the tower AB equals 15 times tan 60°, which simplifies to AB = 15√3.

This calculation is similar to determining how much higher you would reach when standing on a hill. The distance to the top of the tower corresponds to the height of the hill. The steeper the hill (or the greater the angle), the higher the height you could potentially reach.

Hence, the height of the tower is 15√3 m.

After performing the calculations, we conclude that the height of the tower (AB) is 15√3 meters. This means that the actual height of the tower is approximately 25.98 meters when calculated numerically because the value of √3 is about 1.732.

This conclusion can be visualized as accurately measuring how tall a building is compared to how far away you are. It's important because it shows how mathematics can be applied to real-world situations like measuring objects at a distance.

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