9
advanced-math

A medication's concentration in blood follows C(t)=200e0.2tC(t) = 200e^{-0.2t} mg/L where tt is hours. What is the half-life?

A

About 3.5 hours

B

About 5 hours

C

About 2 hours

D

About 7 hours

Correct Answer: A

Choice A is the correct answer. Find when concentration halves.

  1. Half: C(t)=100C(t) = 100 (half of 200).
  2. Equation: 100=200e0.2t100 = 200e^{-0.2t}.
  3. Divide: 0.5=e0.2t0.5 = e^{-0.2t}.
  4. Natural log: ln(0.5)=0.2t\ln(0.5) = -0.2t.
  5. Solve: t=ln(0.5)0.2=0.6930.23.473.5t = \frac{\ln(0.5)}{-0.2} = \frac{-0.693}{-0.2} \approx 3.47 \approx 3.5 hours.

💡 Strategic Tip: Half-life formula: t1/2=ln(2)kt_{1/2} = \frac{\ln(2)}{k} for decay ekte^{-kt}.

Choice B is incorrect because this is too long. Choice C is incorrect because this is too short. Choice D is incorrect because this is way too long.