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Carlos has taken an initial dose of a prescription medication. The relationship between the elapsed time t, in hours, since he took the first dose, and the amount of medication, M(t), in milligrams ( mg ), in his bloodstream is modeled by the following function. M(t)=20*e^(-0.8 t) In how many hours will Carlos have 1mg of medication remaining in his bloodstream? Round your answer, if necessary, to the nearest hundredth. hours

Carlos has taken an initial dose of a prescription medication.\newlineThe relationship between the elapsed time t t , in hours, since he took the first dose, and the amount of medication, M(t) M(t) , in milligrams ( mg \mathrm{mg} ), in his bloodstream is modeled by the following function.\newlineM(t)=20⋅e−0.8t M(t)=20 \cdot e^{-0.8 t} \newlineIn how many hours will Carlos have 1mg 1 \mathrm{mg} of medication remaining in his bloodstream?\newlineRound your answer, if necessary, to the nearest hundredth.\newlinehours

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Q. Carlos has taken an initial dose of a prescription medication.\newlineThe relationship between the elapsed time t t , in hours, since he took the first dose, and the amount of medication, M(t) M(t) , in milligrams ( mg \mathrm{mg} ), in his bloodstream is modeled by the following function.\newlineM(t)=20⋅e−0.8t M(t)=20 \cdot e^{-0.8 t} \newlineIn how many hours will Carlos have 1mg 1 \mathrm{mg} of medication remaining in his bloodstream?\newlineRound your answer, if necessary, to the nearest hundredth.\newlinehours
  1. Given function: Write down the given function that models the amount of medication in Carlos's bloodstream over time.\newlineM(t)=20⋅e(−0.8t)M(t) = 20 \cdot e^{(-0.8t)}
  2. Finding remaining medication time: Set the function equal to 1 mg1\,\text{mg} to find the time when Carlos will have 1 mg1\,\text{mg} of medication remaining in his bloodstream.\newline1=20⋅e(−0.8t)1 = 20 \cdot e^{(-0.8t)}
  3. Isolating the exponential term: Divide both sides of the equation by 2020 to isolate the exponential term.\newline120=e(−0.8t)\frac{1}{20} = e^{(-0.8t)}
  4. Simplifying the equation: Simplify the left side of the equation. 0.05=e(−0.8t)0.05 = e^{(-0.8t)}
  5. Taking the natural logarithm: Take the natural logarithm (ln⁡\ln) of both sides to solve for tt.ln⁡(0.05)=ln⁡(e−0.8t)\ln(0.05) = \ln(e^{-0.8t})
  6. Simplifying the right side: Use the property of logarithms that ln⁡(ex)=x\ln(e^x) = x to simplify the right side of the equation.\newlineln⁡(0.05)=−0.8t\ln(0.05) = -0.8t
  7. Solving for t: Divide both sides by −0.8-0.8 to solve for t.\newlinet=ln⁡(0.05)−0.8t = \frac{\ln(0.05)}{-0.8}
  8. Calculating the value of t: Calculate the value of t using a calculator.\newlinet≈ln⁡(0.05)−0.8t \approx \frac{\ln(0.05)}{-0.8}\newlinet≈−2.9957−0.8t \approx \frac{-2.9957}{-0.8}\newlinet≈3.744625t \approx 3.744625
  9. Rounding the answer: Round the answer to the nearest hundredth. t≈3.74t \approx 3.74 hours

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