Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
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![**Problem Statement:**
A nurse gives a patient 800 mg of medicine at 10:00 a.m. If the half-life of the medicine is 3 hours, how much medicine remains in the patient by 7:00 p.m?
**Explanation:**
To solve this problem, we need to determine the amount of medicine that remains in the patient after a given period using the concept of half-life. The half-life is the time taken for the quantity of a substance to reduce to half its initial amount.
1. **Initial Amount of Medicine:**
- At 10:00 a.m., the patient receives 800 mg of medicine.
2. **Half-Life of Medicine:**
- The half-life of the medicine is given as 3 hours.
3. **Time Elapsed:**
- The time from 10:00 a.m. to 7:00 p.m. is 9 hours.
4. **Number of Half-Life Periods:**
- To find out how many half-life periods have passed, divide the total time elapsed by the half-life duration:
\[
\frac{9 \text{ hours}}{3 \text{ hours/half-life}} = 3 \text{ half-life periods}
\]
5. **Calculation of Remaining Medicine:**
- After each half-life period, the remaining amount of medicine is halved.
- After 1st half-life (3 hours):
\[
\frac{800 \text{ mg}}{2} = 400 \text{ mg}
\]
- After 2nd half-life (6 hours):
\[
\frac{400 \text{ mg}}{2} = 200 \text{ mg}
\]
- After 3rd half-life (9 hours):
\[
\frac{200 \text{ mg}}{2} = 100 \text{ mg}
\]
**Conclusion:**
By 7:00 p.m., 9 hours after the initial administration of the medicine, 100 mg of the medicine remains in the patient.](https://content.bartleby.com/qna-images/question/4f194bc9-7a7c-48af-9b2a-0d49c26655d0/8e60a9ca-7202-4ba6-a63b-14c62fa24497/49nbsnj_processed.jpeg)
Transcribed Image Text:**Problem Statement:**
A nurse gives a patient 800 mg of medicine at 10:00 a.m. If the half-life of the medicine is 3 hours, how much medicine remains in the patient by 7:00 p.m?
**Explanation:**
To solve this problem, we need to determine the amount of medicine that remains in the patient after a given period using the concept of half-life. The half-life is the time taken for the quantity of a substance to reduce to half its initial amount.
1. **Initial Amount of Medicine:**
- At 10:00 a.m., the patient receives 800 mg of medicine.
2. **Half-Life of Medicine:**
- The half-life of the medicine is given as 3 hours.
3. **Time Elapsed:**
- The time from 10:00 a.m. to 7:00 p.m. is 9 hours.
4. **Number of Half-Life Periods:**
- To find out how many half-life periods have passed, divide the total time elapsed by the half-life duration:
\[
\frac{9 \text{ hours}}{3 \text{ hours/half-life}} = 3 \text{ half-life periods}
\]
5. **Calculation of Remaining Medicine:**
- After each half-life period, the remaining amount of medicine is halved.
- After 1st half-life (3 hours):
\[
\frac{800 \text{ mg}}{2} = 400 \text{ mg}
\]
- After 2nd half-life (6 hours):
\[
\frac{400 \text{ mg}}{2} = 200 \text{ mg}
\]
- After 3rd half-life (9 hours):
\[
\frac{200 \text{ mg}}{2} = 100 \text{ mg}
\]
**Conclusion:**
By 7:00 p.m., 9 hours after the initial administration of the medicine, 100 mg of the medicine remains in the patient.
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