A water well drilling rig has dug to a height of -60 ft after one full day of continuous use. a. Assuming the rig drilled at a constant rate, what was the height of the drill after 15 hours?

b. If the rig has been running constantly and is currently at a height of -143.6 ft, for how long has the rig been running? explain

There are 24 hours in a day and so 15 hours is 15÷24=58 of a day. Since the rig drills 60 feet underground in a day, in 58 of a day it will drill

(58 days)×(60feetday)=3008 feet.

So the rig will have drilled 3008=37.5 feet underground in 15 hours.

The rig drills 60 feet underground per day so to find how long it has been running to go 143.6 feet underground, we need to calculate

(143.6 feet)÷(60feetday)=143.660 days.

This is about 2.4 days. Alternatively, it is 57.44 hours

We know that the rig drills at a constant rate, so there is a proportional relationship between the two quantities d, the height to which the drill has dug, and t, the number of days the drill runs. It drills at -60 feet per day, so we can represent this relationship with the equation:

–60t=d

Since 15 hours is 1524=58 days, we can use the equation to find d:  

–60⋅58=d

Since the depth is the same whether we think of it as a positive depth below the surface or a negative height above the surface, we can find this value by multiplying 60⋅58=37.5  and then noting that the sign must be negative if we are representing positions below the surface of the earth by negative numbers. So d=–37.5 and the drill will be at height -37.5 feet after 15 hours.

Likewise, if we know the drill has dug to -143.6 feet, we can use the equation again, this time to find t:  

−60t=−143.6

so t=(−143.6)÷(−60). Since the amount of time would be the same if we were working with positive feet below the surface of the earth and a drill rate of 60 feet per day below the earth surface, we can find this value by dividing 143.6÷60, which is about 2.4 days.