Two buildings 73 meters apart on either side of a highway are shown. From the ninth floor window of the smaller building, Jeffrey looks to the top of the building at an angle of elevation of 31º. He looks to the base of the taller building at an angle of depression of 42º. Find the height of the taller building.
I sketched the situation and saw that
height of tall building = 73tan31° + 73tan42°
= ...
If the 9th floor is x meters up, and the taller building's height is h meters, then if you draw a diagram it should be clear that
h = 73 tan42º + 73 tan31º
To solve this problem, we can use trigonometric functions such as sine, cosine, and tangent.
Let's start by drawing a diagram to visualize the situation.
```
Building A
___________
| |
| |
| |
| | 73 meters
| |
| |
| |
| |
|___________|
|^
| \
| \
h \ Building B
| \
| \
| \
| \
|_______\
```
In the diagram, we have two buildings A and B, which are 73 meters apart. Jeffery is on the 9th floor of building A and is looking at the top of building B at an angle of elevation of 31º. He is also looking at the base of building B at an angle of depression of 42º.
We need to find the height of building B (h).
Let's consider the angle of elevation first.
From the angle of elevation, we can set up a right triangle with the opposite side being the height of building B (h) and the adjacent side being the distance between the two buildings (73 meters). We can use the tangent function to relate the angle and the sides of the triangle:
tan(31º) = h / 73
Next, let's consider the angle of depression.
From the angle of depression, we can set up another right triangle with the opposite side being the height of building B (h) and the adjacent side being the distance between the two buildings (73 meters). However, in this case, the opposite and adjacent sides are switched compared to the previous triangle. Therefore, we need to use the cotangent function (the reciprocal of the tangent function) to relate the angle and the sides of the triangle:
cot(42º) = h / 73
Now we have two equations with the same unknown (h). We can solve this system of equations simultaneously to find the height of building B.
Let's solve these equations:
tan(31º) = h / 73
h = 73 * tan(31º)
h ≈ 41.227 meters
cot(42º) = h / 73
h = 73 * cot(42º)
h ≈ 90.436 meters
Since we have two different values for the height of building B from the two equations, we should take the average to find a more accurate value:
Average height = (41.227 + 90.436) / 2
Average height ≈ 65.831 meters
Therefore, the height of the taller building (building B) is approximately 65.831 meters.