QuestionJuly 15, 2025

Exponential and Logarithmic Models For the following exercises use this scenario: A doctor prescribes 300 milligrams of a therapeutic drug that decays by about 17% each hour. 54. To the nearest minute, what is the half-life of the drug?

Exponential and Logarithmic Models For the following exercises use this scenario: A doctor prescribes 300 milligrams of a therapeutic drug that decays by about 17% each hour. 54. To the nearest minute, what is the half-life of the drug?
Exponential and Logarithmic Models For the following exercises use this scenario: A doctor prescribes 300 milligrams of a therapeutic drug that decays by about 17% each hour. 54. To the nearest minute, what is the half-life of the drug?

Solution
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Answer

Approximately 234 minutes. Explanation 1. Define the decay model The decay model is A(t) = A_0 \cdot e^{kt}, where A_0 is the initial amount, k is the decay rate, and t is time. 2. Calculate the decay rate Given a 17\% decay per hour, k = -0.17. 3. Set up the half-life equation Half-life means A(t) = \frac{A_0}{2}. So, \frac{A_0}{2} = A_0 \cdot e^{-0.17t}. 4. Solve for time t Divide both sides by A_0: \frac{1}{2} = e^{-0.17t}. Take the natural logarithm of both sides: \ln(\frac{1}{2}) = -0.17t. Solve for t: t = \frac{\ln(0.5)}{-0.17}. 5. Convert hours to minutes Calculate t in hours and convert to minutes: t \times 60.

Explanation

1. Define the decay model<br /> The decay model is $A(t) = A_0 \cdot e^{kt}$, where $A_0$ is the initial amount, $k$ is the decay rate, and $t$ is time.<br /><br />2. Calculate the decay rate<br /> Given a $17\%$ decay per hour, $k = -0.17$.<br /><br />3. Set up the half-life equation<br /> Half-life means $A(t) = \frac{A_0}{2}$. So, $\frac{A_0}{2} = A_0 \cdot e^{-0.17t}$.<br /><br />4. Solve for time $t$<br /> Divide both sides by $A_0$: $\frac{1}{2} = e^{-0.17t}$.<br /> Take the natural logarithm of both sides: $\ln(\frac{1}{2}) = -0.17t$.<br /> Solve for $t$: $t = \frac{\ln(0.5)}{-0.17}$.<br /><br />5. Convert hours to minutes<br /> Calculate $t$ in hours and convert to minutes: $t \times 60$.
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