A bridge 100m long is to be built to span a valley as shown. From point a, the angle of depression is 40° and from point b the angle of depression is 71° if a supporting pier is to be constructed (x) find its height to the nearest tenth of a metre
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no diagrams here. You will have to do better describing the relevant points, angles and distances
To find the height of the supporting pier, we can use trigonometry and the given angles of depression.
Let's start by drawing a diagram to visualize the situation:
```
A B
/| |\
/ |h | \
/ | | \
100m | | \
x | | \
|______100m___________|
Bridge
```
In this diagram, A and B represent the points from where the angles of depression are measured. The angle of depression from point A is 40°, and from point B, it is 71°. The bridge has a length of 100m, and we need to find the height (x) of the supporting pier.
Let's consider the right triangles formed by the angles of depression:
- Triangle OPA (where O is the base of the bridge)
- Triangle OPB (where O is the base of the bridge)
For Triangle OPA:
We have the opposite side (height of the pier) and the adjacent side (base of the bridge). We can use the tangent function to relate these sides:
tan(40°) = (height of pier) / 100m
Applying the formula, we can solve for the height of the pier:
(height of pier) = 100m * tan(40°)
Similarly, for Triangle OPB:
We have the opposite side (height of the pier) and the adjacent side (base of the bridge). Again, we can use the tangent function to relate these sides:
tan(71°) = (height of pier) / 100m
Applying the formula, we can solve for the height of the pier:
(height of pier) = 100m * tan(71°)
Now we can calculate the height of the pier:
(height of pier) = 100m * tan(40°) = <<100*tan(40)>> (approx.) meters
(height of pier) = 100m * tan(71°) = <<100*tan(71)>> (approx.) meters
So, the height of the supporting pier to the nearest tenth of a meter is the larger of these values, which is approximately <<max(100*tan(40), 100*tan(71))>> meter(s).