5. The amount Q(t) measured in milligrams of a prescription medica t hours after ingestion is modeled by the equation: Q(t) = 500(.75)* a. What is the rate of decay for this medication? -You body
5. The amount Q(t) measured in milligrams of a prescription medica t hours after ingestion is modeled by the equation: Q(t) = 500(.75)* a. What is the rate of decay for this medication? -You body
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Mathematical Modeling in Pharmacology
#### Problem 5: Medication Decay Over Time
The amount \( Q(t) \) measured in milligrams of a prescription medication that remains in the body \( t \) hours after ingestion is modeled by the equation:
\[ Q(t) = 500 (0.75)^t \]
#### Questions and Solutions:
**a. What is the rate of decay for this medication?**
The decay rate can be determined by understanding the base of the exponential function, \( 0.75 \).
\[ 0.75 = 1 - 0.25 \]
This shows that the rate of decay is \( 0.25 \) or \( 25\% \).
**b. How much of the medication is left after 6 hours? Round to the nearest thousandth.**
To determine the amount left after 6 hours, substitute \( t = 6 \) into the equation:
\[ Q(6) = 500 (0.75)^6 \]
Calculation:
\[ Q(6) = 500 \times 0.177978515625 \approx 88.989 \]
Therefore, approximately 88.989 milligrams of the medication remain after 6 hours.
**c. A second dose can be administered when there are only 80 milligrams remaining. When does this happen? Round to the nearest tenth.**
We need to solve for \( t \) when \( Q(t) = 80 \):
\[ 80 = 500 (0.75)^t \]
Divide both sides by 500:
\[ \frac{80}{500} = (0.75)^t \]
\[ 0.16 = (0.75)^t \]
Taking the logarithm of both sides:
\[ \log(0.16) = t \log(0.75) \]
\[ t = \frac{\log(0.16)}{\log(0.75)} \]
Calculation:
\[ t \approx \frac{-0.796}{-0.125} \approx 6.4 \]
Therefore, a second dose can be administered after approximately 6.4 hours.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc1123ec-a0e0-4e35-afe0-df78f47a58b5%2Fa99e275a-a01d-46fa-a173-128eb331b840%2Fyzb6nnh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Mathematical Modeling in Pharmacology
#### Problem 5: Medication Decay Over Time
The amount \( Q(t) \) measured in milligrams of a prescription medication that remains in the body \( t \) hours after ingestion is modeled by the equation:
\[ Q(t) = 500 (0.75)^t \]
#### Questions and Solutions:
**a. What is the rate of decay for this medication?**
The decay rate can be determined by understanding the base of the exponential function, \( 0.75 \).
\[ 0.75 = 1 - 0.25 \]
This shows that the rate of decay is \( 0.25 \) or \( 25\% \).
**b. How much of the medication is left after 6 hours? Round to the nearest thousandth.**
To determine the amount left after 6 hours, substitute \( t = 6 \) into the equation:
\[ Q(6) = 500 (0.75)^6 \]
Calculation:
\[ Q(6) = 500 \times 0.177978515625 \approx 88.989 \]
Therefore, approximately 88.989 milligrams of the medication remain after 6 hours.
**c. A second dose can be administered when there are only 80 milligrams remaining. When does this happen? Round to the nearest tenth.**
We need to solve for \( t \) when \( Q(t) = 80 \):
\[ 80 = 500 (0.75)^t \]
Divide both sides by 500:
\[ \frac{80}{500} = (0.75)^t \]
\[ 0.16 = (0.75)^t \]
Taking the logarithm of both sides:
\[ \log(0.16) = t \log(0.75) \]
\[ t = \frac{\log(0.16)}{\log(0.75)} \]
Calculation:
\[ t \approx \frac{-0.796}{-0.125} \approx 6.4 \]
Therefore, a second dose can be administered after approximately 6.4 hours.
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