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From Hoffman & Bradley's Calculus for Business, Economics, and the Social and Life Sciences (10th Edition):

Instant coffee is made by adding boiling water (212°F) to coffee mix. If the air temperature is 70°F, Newton's law of cooling states that after $t$ minutes, the temperature will be given by the function $f(t) = 70-A e^{-kt}$. After cooling for 2 minutes, the coffee is still 15° too hot to drink, but 2 minutes the later it is just right. What is this "ideal" temperature for drinking?"

I can determine the value of $A$ but am having trouble with $k4:

$f(t) = 70-Ae^{-kt}$

212 = 70-Ae^0

212 = 70-A

A = -142

f(t) = 70 + 142e^(-kt)

If I let the ideal temperature be called 'y', I am having trouble finding k, as all I (definitively) know in terms of time and temperature is: (0,212) (y+15,2) (y,4).

I do know the average rate of change from t=2 to t=4 was 15 degrees; I don't know how to use this to determine k as I can't take the natural log (ln) of 'y' or 'y+15.'

any help is appreciated.

Thank you kindly.

Rodrigo de Azevedo
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Victor Jaroslaw
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1 Answers1

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So you have

$$ \begin{align*} f(0) &= 70 + 142 \\ f(2) &= 70 + 142 e^{-2k} = y+15 \\ f(4) &= 70 + 142 e^{-4k} = y \end{align*} $$

Subtracting the last two gives

$$ 142 (e^{-2k}-e^{-4k}) = 15 $$

Notice $e^{-4k} = (e^{-2k})^2$, so this is a quadratic equation in $z=e^{-2k}$.

Can you finish from here?

aschepler
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  • I will try, thank you for the – Victor Jaroslaw Sep 29 '22 at 20:17
  • for showing a way forward – Victor Jaroslaw Sep 29 '22 at 20:17
  • Thank you again. I did solve the quadratic by completing the square and got 2 values for z=e^(-2k). Using the 'ln' key, I then obtained 2 separate values for k. I plugged each value for k into f(4) = 70+142e^(-4k). – Victor Jaroslaw Sep 29 '22 at 21:21
  • Yes. One answer is barely above room temperature, which doesn't sound great. The other is close to the ranges found by an Internet search for typical temperatures of hot coffee. – aschepler Sep 29 '22 at 21:29
  • I obtained two temperatures for t=4: 179.95 degrees F, and 72.04 degrees F. Strangely, the answer given in the book (without explanation) was 72.04. Common sense (not the math) dictates that it should be closer to 179 degrees as an ideal temperature; also in our everyday experience, a cup of coffee (usually) does not cool to 72 degrees in 4 minutes. Assuming the book, which seems rigorous is correct, where did I go wrong? The quadratic did provide 2 roots: z =1/2 +/- (41/284)^(1/2) Thank you kindly though for your help, it was excellent! – Victor Jaroslaw Sep 29 '22 at 21:29