4
advanced-math

A coffee cools from 95°C to 59.38°C in 10 minutes. If cooling is exponential with decay factor 0.9 per minute, what is the room temperature (asymptote)?

A

20°C

B

0°C

C

25°C

D

30°C

Correct Answer: A

Choice A is the correct answer. Use the shifted exponential model.

  1. Model: T(t)=(T0Troom)(b)t+TroomT(t) = (T_0 - T_{\text{room}})(b)^t + T_{\text{room}}.
  2. Given: T0=95T_0 = 95, b=0.9b = 0.9, T(10)=59.38T(10) = 59.38.
  3. Set up: 59.38=(95Troom)(0.9)10+Troom59.38 = (95 - T_{\text{room}})(0.9)^{10} + T_{\text{room}}.
  4. Solve: (0.9)100.3487(0.9)^{10} \approx 0.3487, so: 59.38=(95Troom)(0.3487)+Troom59.38 = (95 - T_{\text{room}})(0.3487) + T_{\text{room}}59.38=33.130.3487Troom+Troom59.38 = 33.13 - 0.3487T_{\text{room}} + T_{\text{room}}26.250.6513Troom26.25 \approx 0.6513T_{\text{room}}Troom20°CT_{\text{room}} \approx 20°\text{C}

💡 Strategic Tip: The horizontal asymptote represents room temperature in cooling problems.

Choice B is incorrect because room temperature isn't freezing. Choice C is incorrect because this doesn't match the calculation. Choice D is incorrect because this is too warm.