Pre-Calc

You are taking a road trip in a car without A/C. The temperture in the car is 93 degrees F. You buy a cold pop at a gas station. Its initial temperature is 45 degrees F. The pop's temperature reaches 60 degrees F after 23 minutes.

Given that
\frac{T-A}{T_0 - A} = e^{-kt}
where T = the temperature of the pop at time t.
T_0 = the initial temperature of the pop.
A = the temperature in the car.
k = a constant that corresponds to the warming rate.
and t = the length of time that the pop has been warming up.

How long will it take the pop to reach a temperature of 77 degrees F?

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  1. (T-A)/(To-A) = e^-kt
    We have
    (60-93)/(45-93) = e^(-23k)
    e^(-23k) = .6875
    k = .0163

    So, now we want to find when T=77

    e^(-.0163t) = (77-93)/(45-93) = 0.333
    t = 67.4 minutes

    Nice model, but I'm sure that in real life, after more than an hour, that pop is very nearly 93 degrees!

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