Question

Marc uses a milk frother that needs to be cleaned after use with a drop of dish-washing detergent. He puts a drop of 0.50 gram in the machine and leaves it there for a while. Later Marc rinses the drop from the milk frother by flushing it with water. After many years of experience he has established: that the frother absorbs all the detergent remaining in the milk frother, that the taste of the frothed milk is not affected if there is less than 0.01 gram of detergent in the device, and that, with each flush, two thirds of the detergent is removed. At least how many times does Marc need to flush so as not to taste the detergent? Give your answer in the form of an inequality in n. Write your final answer as n>… where the dots represent the minimal number of times that Marc needs to flush so as not to taste the detergent.

Answer 1

initially 0.5 gram detergent added

after one flush it reduces by two thirds, so it becomes 0.5*(1/3)

after 2 flush it reduces by two thirds two time, so it becomes 0.5*(1/3)*(1/3)

therefore,

aftern flush it becomes 0.5 * (1/3)^n = 0.5 / (3^n)

we want remaining amount less than 0.01 gram

0.5 / (3^n) < 0.01

3^n > 0.5/0.01

3^n > 50

n > log3(50)

n > 3.56

since n is integer : minimum number of flushes = 4

n >= 4

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