Ms hunt is standing on top of the building that is 135m tall. The angle of depression to the top of the building next door is 71 degrees. The angle of depression to the base of the building is 77degrees. The buildings are 31m apart. How tall is the other building
I assumed you made a sketch.
On mine, I labeled the position of Ms Hunt as H on top of the tall building, its base J
For the shorter building, I labeled its top as P and its bottom as Q
I drew a horizontal from H to meet QP extended at R
All kinds of right-angled triangles!
Fill in all your angles.
In triangle HPR,
tan71° = RP/31
RP = 31tan71
in triangle HQR,
tan77° = RQ/31
RQ = 31tan77
then PQ = RQ - RP
=
take over
To find the height of the other building, we can use trigonometry and the concept of angle of depression.
First, let's draw a diagram to better visualize the situation:
```
|\
| \
| \ Ms Hunt
| \
| \
| \ Angle of depression (77°)
| \
--------------- \
| 31m | \
| | \ Height of the building Ms Hunt is standing on (135m)
| | \
| | \
Angle of | | \
depression| Other | \
(71°) | building | \
| | \
| | \
---------------
```
Now, we can solve the problem step by step:
Step 1: Determine the vertical distance from Ms Hunt's position to the base of the building she is standing on.
Using the angle of depression of 77 degrees and the known height of the building of 135m, we can set up the following trigonometric relationship:
tangent(77°) = height / distance
Solving for the height gives us:
height = tangent(77°) * distance
Step 2: Calculate the vertical distance from the top of the building she is standing on to the top of the other building.
We know the 135m height of the building Ms Hunt is standing on and the angle of depression of 71 degrees. With this information, we can use trigonometry again:
tangent(71°) = height / distance
Solving for the height gives us:
height = tangent(71°) * distance
Step 3: Determine the height of the other building.
The height of the other building can be found by subtracting the height from Step 2 (measurement to the top of the other building) from the height from Step 1 (measurement to the base of the building):
height of the other building = height from Step 1 - height from Step 2
Now, let's calculate the values:
Using a scientific calculator, calculate the tangent of 77°:
tangent(77°) ≈ 4.04626
Multiplying the tangent by the distance of 31m, we get:
height from Step 1 ≈ 4.04626 * 31 ≈ 125.71106m
Using a scientific calculator again, calculate the tangent of 71°:
tangent(71°) ≈ 3.07768
Multiplying the tangent by the distance of 31m, we get:
height from Step 2 ≈ 3.07768 * 31 ≈ 95.40608m
Finally, subtracting the height in Step 2 from the height in Step 1, we find:
height of the other building ≈ 125.71106m - 95.40608m ≈ 30.30498m
Therefore, the height of the other building is approximately 30.305m.