Billy is monitoring the exponential decay of a radioactive compound. He has a sample of the compound in a test tube in his lab. According to his calculations, the sample is decaying at a rate of 35% per hour. There are at least 72 grams of the sample remaining. Once the sample reaches a mass of 15 grams, Billy will continually add more of the compound to keep the sample size at a minimum of 15 grams. If R represents the actual amount of the sample remaining, in grams, and t represents the time in hours, then which of the following systems of inequalities can be used to determine the possible mass of the radioactive sample over time?

Solution:Formula for radioactive Decay is given by[tex]R_{0}= R(1-\frac{S}{100})^t[/tex][tex]R_{0}[/tex]= Initial PopulationR = Remaining population after time in hoursRate of Decay = S % per hourInitial Population = 72 gramsFinal population = 15 gramsRate of Decay = 35 % per hourSubstituting the values to get value of t in hours[tex]72=15(1-\frac{35}{100})^t\\\\ 4.8= (0.65)^t\\\\ t= -3.64[/tex]→→1 St expressionBut taking positive value of t , that is after 3.64 hours the sample of 72 grams decays to 15 grams at the rate of 35 % per hour.Now , it is also given that, Once the sample reaches a mass of 15 grams, Billy will continually add more of the compound to keep the sample size at a minimum of 15 grams.Substituting these in Decay FormulaFinal Sample = 15 gmStarting Sample = 15 +k, where k is amount of sample added each time to keep the final sample to 15 grams.Time is over 3.64 hours i.e new time = 3.64 + tRate will remain same i.e 35 % per hour.[tex]15=(15+k)(1-\frac{35}{100})^{3.64+t}[/tex]→→→  Final expression (Second) , that is inequalities can be used to determine the possible mass of the radioactive sample over time.