Question
Marie took 500 mg of Tylenol. The drug depletes in the blood stream at a rate of 20% per hour. The situation can be modeled a(t) = 500(0.8)^t
As t inch, a increases slowly at first then quickly
The y-intercept of the function (0,500)
The range includes all real numbers
As t inch, a increases slowly at first then quickly
The y-intercept of the function (0,500)
The range includes all real numbers
Answers
GPT-4o mini
It looks like you're analyzing the decay of Tylenol (acetaminophen) in the bloodstream based on the model you've provided:
\[ a(t) = 500(0.8)^t \]
This function models the amount of Tylenol left in the bloodstream \( a(t) \) after \( t \) hours. Here's a breakdown of your statements:
1. **Depletion Rate**: You mentioned that the drug depletes in the bloodstream at a rate of 20% per hour. This is reflected in your model since each hour, 80% of the drug remains (hence the factor of \( 0.8 \)).
2. **Behavior of \( a(t) \)**: As \( t \) increases, \( a(t) \) decreases. At first, the decrease may seem slow because the amount is high, but as time progresses and the amount decreases, the changes become more noticeable. This is common in exponential decay functions.
3. **Y-Intercept**: The y-intercept of the function is indeed \( (0, 500) \), which indicates that when \( t = 0 \) (the moment the drug is taken), there is 500 mg of Tylenol present.
4. **Range**: The claim that the range includes all real numbers is inaccurate. In fact, the range of the function \( a(t) \) is \( 0 < a(t) \leq 500 \). Since the amount of Tylenol cannot be negative, the function will approach zero but will never actually reach it.
In conclusion, the function you've described accurately models the decrease of Tylenol in the bloodstream over time, with specific attention to the y-intercept and the range of the function being limited to positive values up to 500.
\[ a(t) = 500(0.8)^t \]
This function models the amount of Tylenol left in the bloodstream \( a(t) \) after \( t \) hours. Here's a breakdown of your statements:
1. **Depletion Rate**: You mentioned that the drug depletes in the bloodstream at a rate of 20% per hour. This is reflected in your model since each hour, 80% of the drug remains (hence the factor of \( 0.8 \)).
2. **Behavior of \( a(t) \)**: As \( t \) increases, \( a(t) \) decreases. At first, the decrease may seem slow because the amount is high, but as time progresses and the amount decreases, the changes become more noticeable. This is common in exponential decay functions.
3. **Y-Intercept**: The y-intercept of the function is indeed \( (0, 500) \), which indicates that when \( t = 0 \) (the moment the drug is taken), there is 500 mg of Tylenol present.
4. **Range**: The claim that the range includes all real numbers is inaccurate. In fact, the range of the function \( a(t) \) is \( 0 < a(t) \leq 500 \). Since the amount of Tylenol cannot be negative, the function will approach zero but will never actually reach it.
In conclusion, the function you've described accurately models the decrease of Tylenol in the bloodstream over time, with specific attention to the y-intercept and the range of the function being limited to positive values up to 500.