On the ceiling of a cavern, a stalactite grows downward .004 inches per year. ten feet directly below on the cavern’s floor, a stalagmite grows upward .006 inches per year. how many years will it take them to meet?

The stalactites become 0.004 + 0.006 = 0.01 inches closer each year, and begin 10 feet apart, or 120 inches apart. 

let y be the distance between them, and x be the number of years that go by:
y = -0.01x + 120

We want to find where y = 0, where they meet:
0 = -0.01x + 120        Add 0.01x to both sides
0.01x = 120               Multiply both sides by 100
x = 12,000 

The stalagmite and stalactite will meet in 12,000 years