The amount of a certain medication in a patient's body over time is given by the (t) 500(0.93) where A represents the amount of medication (milligrams) and t is the time (hours). How many hours will it be before the doctor must administer the medication again if it should be repeated when the level of medication in the patient decreases to 100 mg? Show your work using the Equation Editor tool. equation At Paragraph = BI U A EV O
The amount of a certain medication in a patient's body over time is given by the (t) 500(0.93) where A represents the amount of medication (milligrams) and t is the time (hours). How many hours will it be before the doctor must administer the medication again if it should be repeated when the level of medication in the patient decreases to 100 mg? Show your work using the Equation Editor tool. equation At Paragraph = BI U A EV O
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter7: Exponents And Exponential Functions
Section7.8: Transforming Exponential Expressions
Problem 1CYU
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![## Medication Decay Over Time
The amount of a certain medication in a patient's body over time is given by the equation:
\[ A(t) = 500(0.93)^t \]
where \( A \) represents the amount of medication (in milligrams) and \( t \) is the time (in hours).
### Problem Statement
How many hours will it be before the doctor must administer the medication again if it should be repeated when the level of medication in the patient decreases to 100 mg? Show your work using the Equation Editor tool.
### Solution
To determine the time \( t \) when the medication level decreases to 100 mg, we need to solve the equation:
\[ 100 = 500(0.93)^t \]
1. **Isolate the exponential term**:
\[ \frac{100}{500} = (0.93)^t \]
\[ 0.2 = (0.93)^t \]
2. **Take the natural logarithm of both sides**:
\[ \ln(0.2) = \ln((0.93)^t) \]
\[ \ln(0.2) = t \cdot \ln(0.93) \]
3. **Solve for \( t \)**:
\[ t = \frac{\ln(0.2)}{\ln(0.93)} \]
4. **Compute the value**:
\[ t \approx \frac{-1.6094}{-0.0741} \]
\[ t \approx 21.72 \]
Therefore, the doctor should administer the medication again after approximately 21.72 hours.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3692e55-09d2-4c7f-90e9-0ed90d8b0dce%2Fced7e219-1acb-4a0e-aea2-2072c36200c9%2Fbrhawg7_processed.png&w=3840&q=75)
Transcribed Image Text:## Medication Decay Over Time
The amount of a certain medication in a patient's body over time is given by the equation:
\[ A(t) = 500(0.93)^t \]
where \( A \) represents the amount of medication (in milligrams) and \( t \) is the time (in hours).
### Problem Statement
How many hours will it be before the doctor must administer the medication again if it should be repeated when the level of medication in the patient decreases to 100 mg? Show your work using the Equation Editor tool.
### Solution
To determine the time \( t \) when the medication level decreases to 100 mg, we need to solve the equation:
\[ 100 = 500(0.93)^t \]
1. **Isolate the exponential term**:
\[ \frac{100}{500} = (0.93)^t \]
\[ 0.2 = (0.93)^t \]
2. **Take the natural logarithm of both sides**:
\[ \ln(0.2) = \ln((0.93)^t) \]
\[ \ln(0.2) = t \cdot \ln(0.93) \]
3. **Solve for \( t \)**:
\[ t = \frac{\ln(0.2)}{\ln(0.93)} \]
4. **Compute the value**:
\[ t \approx \frac{-1.6094}{-0.0741} \]
\[ t \approx 21.72 \]
Therefore, the doctor should administer the medication again after approximately 21.72 hours.
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