Coffee Cooling and the Limits of an Exponential Model
A cup of coffee is poured at 95 degrees Celsius into a room kept at 20 degrees Celsius. A student models the coffee's temperature T(t) in degrees Celsius after t minutes with T(t)=20+75(0.9)^t, reasoning that each minute the coffee loses 10% of its excess temperature above room temperature. The student wants to drink the coffee when it is at most 40 degrees Celsius, but also does not want to wait so long that it becomes cold. A classmate claims that after 30 minutes the coffee will be exactly at room temperature because 0.9^30 is very small, and that the model eventually predicts a temperature below 20 degrees Celsius. The student also wonders whether the 10% per minute assumption is reasonable, given that the temperature drop in the first minute is larger than the drop in the tenth minute.
1. Use the model to estimate the coffee's temperature after 10 minutes.
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Substitute t=10: T(10) = 20 + 75(0.9)^10. Now (0.9)^10 is about 0.34868, so 75(0.34868) is about 26.15. Thus T(10) is about 20 + 26.15 = 46.15 degrees Celsius, or about 46.2 degrees Celsius. The coffee is still above the 40-degree drinking threshold after 10 minutes.
2. Solve for the time when the coffee first reaches 40 degrees Celsius.
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Set 20 + 75(0.9)^t <= 40. Then 75(0.9)^t <= 20, so (0.9)^t <= 20/75 = 4/15. Taking natural logarithms gives t ln(0.9) <= ln(4/15). Since ln(0.9) is negative, dividing reverses the inequality: t >= ln(4/15)/ln(0.9). Numerically, ln(4/15) is about -1.3218 and ln(0.9) is about -0.10536, so t is about 12.55 minutes. The coffee first reaches 40 degrees Celsius after about 12.5 minutes, and it continues to cool below that temperature afterward.
3. Evaluate the classmate's claims about the 30-minute temperature and about eventually dropping below room temperature.
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At t=30, T(30) = 20 + 75(0.9)^30. Since (0.9)^30 is about 0.04239, the excess is about 3.18 degrees, so T(30) is about 23.2 degrees Celsius, not 20. The model has horizontal asymptote T=20 because 75(0.9)^t is always positive for finite t. It approaches room temperature from above but never equals or falls below 20 degrees Celsius. The classmate confuses a small exponential factor with zero. The 10% assumption also implies that the absolute temperature drop shrinks as the excess shrinks, which is why the first-minute drop is larger than the tenth-minute drop.