Suppose that the level of medicine in the patient's body must maintain a level greater than 100 milligrams.

Question

After how many hours will this patient first need to take the additional medicine?
Show your work or explain your answer to receive credit for this question.

Answer 1

To determine when the patient will first need to take additional medicine, we can use the given information. Let's suppose the medicine is being metabolized at a rate of 10% per hour.

After 1 hour: 100 mg * 0.10 = 10 mg is metabolized
Remaining medicine = 100 mg - 10 mg = 90 mg

After 2 hours: 90 mg * 0.10 = 9 mg is metabolized
Remaining medicine = 90 mg - 9 mg = 81 mg

After 3 hours: 81 mg * 0.10 = 8.1 mg is metabolized
Remaining medicine = 81 mg - 8.1 mg = 72.9 mg

After 4 hours: 72.9 mg * 0.10 = 7.29 mg is metabolized
Remaining medicine = 72.9 mg - 7.29 mg = 65.61 mg

After 5 hours: 65.61 mg * 0.10 = 6.561 mg is metabolized
Remaining medicine = 65.61 mg - 6.561 mg = 59.049 mg

After 6 hours: 59.049 mg * 0.10 = 5.9049 mg is metabolized
Remaining medicine = 59.049 mg - 5.9049 mg = 53.1441 mg

After 7 hours: 53.1441 mg * 0.10 = 5.31441 mg is metabolized
Remaining medicine = 53.1441 mg - 5.31441 mg = 47.82969 mg

After 8 hours: 47.82969 mg * 0.10 = 4.782969 mg is metabolized
Remaining medicine = 47.82969 mg - 4.782969 mg = 43.046721 mg

After 9 hours: 43.046721 mg * 0.10 = 4.3046721 mg is metabolized
Remaining medicine = 43.046721 mg - 4.3046721 mg = 38.7420489 mg

After 10 hours: 38.7420489 mg * 0.10 = 3.87420489 mg is metabolized
Remaining medicine = 38.7420489 mg - 3.87420489 mg = 34.86784401 mg

After 11 hours: 34.86784401 mg * 0.10 = 3.486784401 mg is metabolized
Remaining medicine = 34.86784401 mg - 3.486784401 = 31.381059609 mg

After 12 hours: 31.381059609 mg * 0.10 = 3.1381059609 mg is metabolized
Remaining medicine = 31.381059609 mg - 3.1381059609 mg = 28.2429536481 mg

After 13 hours: 28.2429536481 mg * 0.10 = 2.82429536481 mg is metabolized
Remaining medicine = 28.2429536481 mg - 2.82429536481 mg = 25.41865828329 mg

After 14 hours: 25.41865828329 mg * 0.10 = 2.54186582833 mg is metabolized
Remaining medicine = 25.41865828329 mg - 2.54186582833 mg = 22.876792454959 mg

After 15 hours: 22.876792454959 mg * 0.10 = 2.2876792454959 mg is metabolized
Remaining medicine = 22.876792454959 mg - 2.2876792454959 mg = 20.5891132094631 mg

After 16 hours: 20.5891132094631 mg * 0.10 = 2.05891132094631 mg is metabolized
Remaining medicine = 20.5891132094631 mg - 2.05891132094631 mg = 18.5302018885168 mg

After 17 hours: 18.5302018885168 mg * 0.10 = 1.85302018885168 mg is metabolized
Remaining medicine = 18.5302018885168 mg - 1.85302018885168 = 16.6771816996651 mg

So, after 17 hours, the patient's medicine level drops below 100 mg and they will need to take additional medicine at that point.