2
advanced-math

Coffee at 180°F cools in a 70°F room. After 5 minutes it's 150°F. Using Newton's Law T(t)=70+110ektT(t) = 70 + 110e^{-kt}, find kk.

A

About 0.063

B

About 0.1

C

About 0.05

D

About 0.15

Correct Answer: A

Choice A is the correct answer. Solve for the cooling constant kk.

  1. Given: T(5)=150T(5) = 150, so 150=70+110e5k150 = 70 + 110e^{-5k}.
  2. Isolate: 80=110e5k80 = 110e^{-5k}.
  3. Divide: e5k=80110=8110.727e^{-5k} = \frac{80}{110} = \frac{8}{11} \approx 0.727.
  4. Natural log: 5k=ln(0.727)0.318-5k = \ln(0.727) \approx -0.318.
  5. Solve: k=0.31850.0640.063k = \frac{0.318}{5} \approx 0.064 \approx 0.063.

💡 Strategic Tip: Use given data points to find unknown constants in exponential models.

Choice B is incorrect because this cooling rate is too fast. Choice C is incorrect because this is slightly too slow. Choice D is incorrect because this rate is way too fast.