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

= Initial Population

R = Remaining population after time in hours

Rate of Decay = S % per hour

Initial Population = 72 grams

Final population = 15 grams

Rate of Decay = 35 % per hour

Substituting the values to get value of t in hours

→→1 St expression

But 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 Formula

Final Sample = 15 gm

Starting 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 + t

Rate will remain same i.e 35 % per hour.

→→→  Final expression (Second) , that is inequalities can be used to determine the possible mass of the radioactive sample over time.

The required system of inequalities is, and

Step-by-step explanation:

We are given that,

The actual amount of sample remaining (in grams) = R.

Time (in hours) = t

The formula for the radioactive decay given by, , where k = decay rate

As there are initially at-least 72 grams of the sample, which is decreasing at the rate of 35% = 0.35.

So, and k = 0.35

Thus, we get by substituting the values in the formula above,

Moreover, the minimum size of the sample is 15 grams. So, we have,

Hence, the required system of inequalities is,

correct

step-by-step explanation: