Many treatments for diseases and other conditions require a patient to maintain a certain concentration of a drug in their body. Since the body continuously metabolizes and excretes the drug, regular doses are needed to maintain the appropriate concentration. Care must be taken to be sure that the concentration remains within a particular range. If the concentration is too low, the drug will be ineffective; if the concentration is too high, the drug may have harmful or fatal side effects. After a single dose of a drug is taken, the amount of drug in a patient's body follows exponential decay: A = De-kt, where the constant D is the dose amount, and the variable t is the time in hours. If a patient takes regular doses of the drug, we can find the amount of drug remaining in the patient's body immediately before taking each dose. Let s represent the time (in hours) between doses. Now the amount of drug remaining in the patient's body immediately before taking the second dose is R1 = De-ks. The amount drug remaining in the patient's body immediately before taking the third dose is R2 = De-ks + De-2ks; in this expression, the first term represents the amount of drug remaining from the second dose (which was taken s hours ago), and the second term represents the amount of drug remaining from the first dose (which was taken 2s hours ago). Continuing this process for n steps, the amount of drug remaining in the patient's body immediately before taking the (n + 1)th dose is Rn = De-ks + De-2ks + De-3ks + . . . + De-nks

1-Find the Decay Constant (2 points) The half-life of a specific drug in the body is 27.5 hours. Use this information to solve for the decay constant k. Round this value to 4 places.

2-Notice that the expression for Rn is a geometric series! What is the common ratio for this series? Use the decay constant you found in the previous step along with a dosage interval of s = 24 hours. Round this value to 2 places.

3-After many doses, the amount of drug remaining in a patient's body reaches equilibrium; in other words, the amount of drug remaining in the patient's body immediately before each dose is nearly the same as the amount remaining immediately before the next dose. This amount can be found by finding the sum of the infinite geometric series R = De-ks + De-2ks + De-3ks + . . . A patient is directed to take a 140 mg dose of the drug every 24 hours. Use these values along with the decay constant you found earlier to find the equilibrium amount of drug in the patient's body immediately before each dose after many doses. Round this value to 2 places

4-In order to be effective, the minimum drug concentration at equilibrium (amount of drug immediately before a dose / mass of patient) must be at least 2 mg/kg. In order to be safe, the maximum drug concentration at equilibrium (amount of drug immediately after a dose / mass of patient) must not exceed 3.5 mg/kg. Is it effective and safe for an 80 kg patient to take a 140 mg dose of the drug every 24 hours?

5-Instead of taking a 140 mg dose every 24 hours, the 80 kg patient is now directed to take a 70 mg dose every 12 hours. Redo the previous three steps (Common Ratio, Equilibrium, Effective and Safe) for the new treatment plan. Is this treatment effective and safe?