Estabon poured himself a hot beverage that had a temperature of 198 then set it on the kitchen table to cool. The temperature of the kitchen was a constant 68. Of the drink cooled to 182 in five minutes how long will it take for the drink to cool to 90?
Simple sol'n: Rate of decrease in temperature: -16/5 °C/min 182 + x(-3.2) = 90 -3.2x = -92 x = 28.75 min's Total time to cool to 90 °C = 28.75 + 5 = 33.75 min's
By Differential Equations: dT/dt = -kT, t = time, T = temperature of beverage If you don't understand what I've written above, it is simply: 'The change in T with respect to t is proportional to T' or 'The rate of cooling is proportional to T' Continuing: Separating the variables: 1/T dT = -k dt
Integrate both sides: ∫1/T dT = ∫-k dt ln(T) = -kt + c
From what we're given: when t = 0, T = 198 ln(198) = -k(0) + c c = ln(198) so... ln(T) = -kt + ln(198)
