From a point B that is 9 meters above level ground , the angle of elevation of the top of a building is 43 degrees and the angle of depression of the base of the building is 7 degrees. What is the height of the building.
(tell what formula and why you used it will very much be appreciated)
draw a horizontal line from B to where it intersects the building wall. Call that point C.
Label the top of the building A, and the bottom D.
The height of the building is thus
AD = AC + CD
AC/BC = tan 43°
CD/BC = tan 7°
Now, you see that without knowing the distance BC, there is no way to determine the height of the building. The farther away you are, the higher the building.
To find the height of the building, we can use the trigonometric concept of tangent.
The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle in a right triangle.
Let's visualize this problem using a diagram:
```
B
/|
/ | height of the building
/ |
/ |
/θa |
/_____|______
9m x <- ground level
base of the
building
```
In this diagram, B is the point above the building, and θa is the angle of elevation. We know that the angle of elevation is the angle between the horizontal ground and the line of sight to the top of the building.
Now, let's find the value of the tangent of the angle of elevation:
tan(θa) = opposite / adjacent
tan(θa) = height of the building / base of the building
Rearranging the equation, we get:
height of the building = base of the building * tan(θa)
Substituting the given values, we have:
height of the building = 9m * tan(43°)
Using a scientific calculator or a trigonometric table, we can find the value of tan(43°), which is approximately 0.932.
Thus:
height of the building = 9m * 0.932
height of the building ≈ 8.388 meters
Therefore, the height of the building is approximately 8.388 meters.