## Medication Decay Over TimeThe 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 StatementHow 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.### SolutionTo 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.

The amount of certain medication in a patient body over time is given by the equation Where A…

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

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

## 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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