How much contaminant is removed during the first three hours?

Question

Many physical processes are exponential in nature. A typical way of cleaning a tank of contaminated water is to run in clean water at a constant rate, stir, and let the mixture run out at the same rate. Suppose there are initially 100 pounds of a contaminant in a large tank of water. Assume that the cleaning method described above removes 10% of the remaining contaminant each hour.

How much contaminant is removed during the first three hours?

Answer 1

The quick version of this answer is that 10% of 100 pounds is removed, then 10% removed from whatever remains, then 10% removed again.

 

That means that after one hour,

the amount remaining will be 100×90%

 

After two hours,

the amount remaining will be 100×90%×90%

 

After three hours,

the amount remaining will be 100×90%×90%×90%

 

Do you see the pattern?

 

Repeated multiplication is written as exponents, 

so after n hours,

the amount remaining will be 100(90%)ⁿ