9.1.6. Example 6

The angles of depression of the top and the bottom of an 8 m tall building from the top of a multi-storeyed building are 30° and 45°, respectively. Find the height of the multistoreyed building and the distance between the two buildings.

In this problem, we are given a scenario involving two buildings: an 8-meter tall building and a multi-storeyed building. We need to determine the height of the multi-storeyed building and the horizontal distance between the two buildings based on the angles of depression measured from the top of the taller building.

Imagine standing on the roof of a tall building and looking down at a shorter building. The angle at which you look down to see the top of the shorter building is one angle of depression, while the angle to see the bottom is another. These angles help us figure out not just how high the building is, but also how far away it is.

In the right triangle formed, the angle at point PBD, which is the point of observation at the top of the multi-storeyed building, corresponds to the angle of depression to the top of the 8m building that is 30°. Similarly, the angle PAC (the angle to the bottom) is 45°.

The angles of depression are important because they are related to the angles in the triangles we create when we draw lines from the observer to the points on the lower building. The angle of 30° gives us one triangle, and 45° gives us another triangle. This allows us to set up equations based on the properties of these triangles.

Think of a slide at a playground. The angle you slide down is like the angle of depression. If the slide is steep, like 30°, you'd go down quickly (indicating that the bottom of the building is further away), but if it’s more gradual, like 45°, you would go down slower, showing that the building is closer.

In triangle PBD, we have PD/BD = tan 30° = 1/sqrt(3) or BD = PD * sqrt(3). In triangle PAC, the relationship is given by PC/AC = tan 45° = 1, which leads to PC = AC.

We apply trigonometric ratios to relate the angles we have to the sides of our triangles. For the triangle with angle 30°, we find the relation between one side (PD) and the opposite side (BD). For the triangle with angle 45°, since tan 45° equals 1, we know the two sides are equal, leading us to the relationship PC = AC.

Think about using a ladder against a wall: If you know the angle of the ladder and the height you want to reach, you can use trigonometry to find out how far the base of the ladder needs to be placed away from the wall.

We conclude that PC = PD + DC, and from our earlier findings, we know PD + 8 = BD = PD * sqrt(3).

This section combines the height of the taller building and the height relationships we've established through trigonometry. We set up the equation based on the relationships of PD, the height of the two buildings, and the distances derived earlier.

Imagine two people building towers of blocks. The first person builds a tower of 8 blocks high while the second person has an unknown number above that. The total height of the second tower needs to equal the first tower plus some extra blocks. Similarly, we find the total height of the multi-storeyed building by figuring out these component heights.

From the earlier relationship, we derive PD = (8 * (3 - sqrt(3))) / (3 - 1). Thus, after calculations, we find the height of the multi-storeyed building is 8 + PD and the distance AC is also calculated as 4 * sqrt(3) m.

At this point, we solve our equation to find the height of the multi-storeyed building. By determining PD and then adding the 8 m height of the smaller building, we finally find out how tall the larger building is and the specific distance separating them.

Think of it like measuring the height of a tree you cannot directly measure. You see the top from a distance and use your measuring tools (the angles) to compute both how tall it is and how far away you are standing. In this case, through calculation, we can determine the 'height' of our invisible tree (the multi-storeyed building)!

--