A team of doctors reports in a medical journal that a drug is metabolized at the rate y = 175(0.994)^t, where t is the time in minutes and y is the amount of the drug remaining in... A team of doctors reports in a medical journal that a drug is metabolized at the rate y = 175(0.994)^t, where t is the time in minutes and y is the amount of the drug remaining in the bloodstream in ppm. Interpret the meaning of 175 and 0.994 in the context of this situation. Does more than half the drug remain after 4 hours? Justify your answer.
Understand the Problem
The question is asking for interpretations of specific values in a mathematical model of drug metabolism. It also requires evaluating whether more than half of the drug remains after a certain time period, specifically 4 hours, and requires justification for the answers.
Answer
No, more than half the drug does not remain after 4 hours; approximately 23.73 ppm remains, which is less than 87.5 ppm.
Answer for screen readers
More than half the drug does not remain after 4 hours.
Steps to Solve
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Interpret 175 in context The value 175 represents the initial amount of the drug in the bloodstream at time $t = 0$ minutes. It indicates that when no time has passed, there are 175 ppm (parts per million) of the drug present.
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Interpret 0.994 in context The value 0.994 is the decay factor. It shows the rate at which the drug is metabolized. Specifically, for every minute that passes, 99.4% of the drug remains. This means that 0.6% of the drug is metabolized every minute.
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Calculate the remaining amount after 4 hours First, convert 4 hours to minutes: $$ t = 4 \times 60 = 240 \text{ minutes}. $$ Now substitute $t$ into the equation: $$ y = 175(0.994)^{240}. $$ Calculate $(0.994)^{240}$ first.
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Calculate $(0.994)^{240}$ Using a calculator or computational software, we find: $$ (0.994)^{240} \approx 0.1356. $$
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Final calculation of remaining drug amount Now substitute this back into the equation: $$ y = 175 \times 0.1356 \approx 23.73 \text{ ppm}. $$
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Determine if more than half remains Half of the initial amount (175 ppm) is: $$ \frac{175}{2} = 87.5 \text{ ppm}. $$ Since 23.73 ppm is less than 87.5 ppm, the answer is no.
More than half the drug does not remain after 4 hours.
More Information
The initial amount of the drug in the bloodstream is 175 ppm, and after 4 hours, only about 23.73 ppm remains, which is significantly less than half of the initial amount. This rapid decline shows the effectiveness of the metabolism process.
Tips
- A common mistake is miscalculating the decay factor $(0.994)^{240}$. Ensure correct use of a calculator for exponentiation.
- Another mistake is not converting hours to minutes properly, which is crucial for correctly applying the formula.
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