9.1.5. Example 5

The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun’s altitude is 30° than when it is 60°. Find the height of the tower.

In this example, we have a tower whose shadow changes length depending on the position of the sun in the sky. When the sun is at an angle of 60°, the shadow is shorter compared to when the sun is at an angle of 30°. We know that at 30°, the shadow is 40 meters longer than at 60°. The goal is to determine the height of the tower based on these conditions.

Imagine standing next to a streetlamp at different times of the day. In the early morning (like the 30° angle), your shadow is long. But when the sun is directly above at noon (similar to 60°), your shadow is much shorter. The difference in shadow lengths can help you understand how tall the streetlamp is.

Let AB be h m and BC be x m. According to the question, DB is 40 m longer than BC. So, DB = (40 + x) m.

To solve the problem, we need to define the variables. Let 'h' be the height of the tower (AB) and 'x' be the length of the shadow when the sun's altitude is 60° (BC). The problem states that when the sun's altitude is 30°, the new shadow length (DB) is 40 meters longer than the shadow at 60°, meaning DB = x + 40.

Think of measuring the height of a tree. You decide to use two sticks. One stick (x) represents a shorter shadow when the sun is high, and the other one (DB) represents a longer shadow when the sun is lower. The difference in their lengths helps you understand how much taller the tree is.

Now, we have two right triangles ABC and ABD. In △ABC, tan 60° = AB/BC or, 3 = h/x (1). In △ABD, tan 30° = AB/DB i.e., 1/√3 = h/(x + 40) (2).

The shadows and the height of the tower create two right triangles. In triangle ABC, the angle of elevation is 60°, and we can express the tangent of that angle as the height of the tower divided by the length of the shadow (tan(60°) = h/x). Similarly, in triangle ABD, at an angle of 30°, we can set up another equation using tangent (tan(30°) = h/(x + 40)). The tangent values correspond to known trigonometric ratios.

If you were to measure how tall a building is using a clinometer, the angles of elevation correspond to your line of sight. The angles give you two different triangles, and by comparing the triangles, you can use trigonometry to find out how tall the building is.

From (1), we have h = 3x. Putting this value in (2), we get (3x)√3 = x + 40, i.e., 3x = x + 40, i.e., x = 20.

After establishing our equations, we can isolate 'h' in the first equation to find h = 3x. We then substitute this expression for 'h' into the second equation. This allows us to solve for 'x'. After calculations, we find that x equals 20 meters, which represents the shadow's length when the sun's angle is 60°.

Imagine you have a recipe that requires 3 cups of flour for every 1 cup of sugar. If you need to adjust your recipe based on how much sugar you have (similarly to how we adjust our equations), you can find out the exact measurements needed to keep the proportions correct.

So, h = 20√3 [From (1)]. Therefore, the height of the tower is 20√3 m.

Having calculated 'x', we can now substitute it back into our expression for 'h'. We find that the height of the tower is 20 times the square root of 3, which gives us the full height in meters.

If you were to find out how tall a building is through calculations based on different shadows it casts, this final calculation brings you the answer, just like measuring the final height after determining how much material you need for construction.

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