Newton's Law of Cooling

2
advanced-math

A hot object at 200°F is placed in a room at 70°F. The temperature T(t)T(t) follows Newton's Law of Cooling: T(t)=70+130e0.1tT(t) = 70 + 130e^{-0.1t} where tt is in minutes. What is the temperature after 10 minutes?

A

About 118°F

B

About 150°F

C

About 70°F

D

About 85°F

Correct Answer: A

Choice A is the correct answer. Evaluate at t=10t = 10.

  1. Substitute: T(10)=70+130e0.1(10)=70+130e1T(10) = 70 + 130e^{-0.1(10)} = 70 + 130e^{-1}.
  2. Calculate: e1=1e0.3679e^{-1} = \frac{1}{e} \approx 0.3679.
  3. Compute: 130(0.3679)47.83130(0.3679) \approx 47.83.
  4. Result: T(10)=70+47.83118°FT(10) = 70 + 47.83 \approx 118°F.

💡 Strategic Tip: Newton's Law: T(t)=Troom+(T0Troom)ektT(t) = T_{\text{room}} + (T_0 - T_{\text{room}})e^{-kt}.

Choice B is incorrect because this doesn't account for exponential decay. Choice C is incorrect because it hasn't cooled to room temperature yet. Choice D is incorrect because the decay is not that rapid.