6
advanced-math

A chemical reaction follows C(t)=500e0.15tC(t) = 500e^{-0.15t} grams where tt is minutes. When is the concentration 100 grams?

A

About 10.7 minutes

B

About 15 minutes

C

About 5 minutes

D

About 20 minutes

Correct Answer: A

Choice A is the correct answer. Solve for tt when C(t)=100C(t) = 100.

  1. Equation: 100=500e0.15t100 = 500e^{-0.15t}.
  2. Divide: 0.2=e0.15t0.2 = e^{-0.15t}.
  3. Natural log: ln(0.2)=0.15t\ln(0.2) = -0.15t.
  4. Solve: t=ln(0.2)0.15=1.6090.1510.7310.7t = \frac{\ln(0.2)}{-0.15} = \frac{-1.609}{-0.15} \approx 10.73 \approx 10.7 minutes.

💡 Strategic Tip: For decay to reach 1/5 of original: ln(0.2)1.609\ln(0.2) \approx -1.609.

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