A deep mountain valley is separated by two vertical cliffs. One cliff is 600 meters high and the other is 400 meters high. A cable car runs from the foot of each cliff to the top of the other cliff. At what height above the ground do the two cable cars cross? Be sure to draw a diagram.
Let L be the distance between cliffs and y be the height where the cables intersect, x is the horizontal distance from the lower cliff to below the intersection point.
y/x = 600/L
y/(L-x) = 400/L
(L-x)/y * (y/x) = 600/400 = 3/2
(L-x)/x = 3/2
L/x -1 = 3/2
L/x = 5/2
x/L = 2/5
y = (x/L)*600 = 240 m
You will have to draw your own figure.
To find the height at which the two cable cars cross, we need to consider the altitude at that point. Let's assume that the height at which they cross is 'h' meters above the ground.
Now, let's draw a diagram to visualize the situation:
```
Cable Car A
|
---------------------- ^
| /\ | \ |
| / \ | \ |
| / \ | \ |
| / \ | \ |
|/ \ | \ | h
/----------\ | \ |
/ \ | \ |
Cliff A (600m) / \ | \ |
/ \ | \ |
/ \| \ |
/ /\ \/
/ / \ -----
/ Cliff B \ | h |
-------------------------- -----
Cable Car B
```
In this diagram, Cable Car A is traveling from the foot of Cliff A to the top of Cliff B, and Cable Car B is traveling from the foot of Cliff B to the top of Cliff A. We are looking for the height at which they cross, represented by 'h' meters above the ground.
To solve for 'h', we can use the concept of similar triangles. If we consider the triangle formed by Cable Car A, the height 'h', and the height of Cliff A (600m), and also the triangle formed by Cable Car B, the height 'h', and the height of Cliff B (400m), we can set up the following proportion:
h / 600 = h / 400
To solve this proportion, we can cross-multiply:
400h = 600h
And then simplify:
600h - 400h = 0
200h = 0
Therefore, from this equation, we can see that 'h' can be any value as long as it satisfies the equation. In other words, the two cable cars can cross at any height above the ground, as long as they are crossing at the same altitude.
Therefore, the height at which the two cable cars cross can be at any arbitrary value.