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.
height h
I assume cables are straight
660 on left, 400 on right
distance x on left, y on right from cross point to cliff on ground
x/h = (x+y)/400
y/h = (x+y)/600
h = 400 x/(x+y)
h = 600 y/(x+y)
so 400 x = 600 y
x = 1.5 y
total distance = 2.5 y
2.5 y/600 = y/h
2.5 h = 600
h = 240
To solve this problem, we can draw a diagram to visualize the situation.
Let's assume the left cliff is 600 meters high and the right cliff is 400 meters high. We will represent the heights using vertical lines.
```
|
|
| Cliff 1 (600m)
| |----|
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| |----|
|
| | |
| Cliff 2 (400m)
|
|-------------------------------
Ground level
```
Next, we need to find the point at which the two cable cars cross each other. We'll call this point "X".
```
|
|
| Cliff 1 (600m)
| |----|
| | |
| | |
| | |
| | |
| | |
| | X |
| | |
| | |
| | |
| | |
| |----|
|
| | |
| Cliff 2 (400m)
|
|-------------------------------
Ground level
```
Since both cable cars start at the foot of the cliffs, they must meet at the same height from the ground level. Let's call this height "h".
```
|
| |
| Cliff 1 (600m)
| | |
| | |
| | |
| | |
| | |
| | X |
| | |
| | |
| | |
| | |
| |----|
|
| | |
| Cliff 2 (400m)
|
|-------------------------------
Ground level
```
To determine the height "h", we can use the similar triangles formed by the two cable cars and the cliffs.
Let's label the height from the ground level to the meeting point of the cable cars as "h".
```
|
| |
| Cliff 1 (600m)
| | |
| | |
| | |
| | |
| | |
| | X |
| | |
| | |
| | |
| | |
| |----|
|
| | |
| Cliff 2 (400m)
|
|-------------------------------
Ground level
```
From the diagram, we can see that the ratio of the height on Cliff 1 to the height on Cliff 2 is the same as the ratio of the distance from the ground level to the meeting point on Cliff 1 to the distance from the ground level to the meeting point on Cliff 2.
In other words,
```
600 / 400 = (600 - h) / h
```
Now we can solve this equation to find the value of "h".
600 / 400 = (600 - h) / h
1.5 = (600 - h) / h
Cross-multiplying:
1.5h = 600 - h
Combining like terms:
1.5h + h = 600
2.5h = 600
Dividing by 2.5 on both sides:
h = 600 / 2.5
h = 240
Therefore, the two cable cars cross each other at a height of 240 meters above the ground level.
I hope this explanation helps you understand how to solve this problem!