Exponential Problem

**Macleef** · January 24th 2009, 05:05 PM

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

$212 = 70 - Ae^{0}$

$A = 212 - 70$

$A = 142$

$15 = 70 - 142e^{-k2}$

$ln{\frac{\frac{(15-70)}{-142}}{-2}} = k$

$0.47 = k$

$f(4) = 70 - 142e^{-0.47\times4}$

$f(4) = 48.697F$

...is my answer correct?

**Chris L T521** · January 24th 2009, 05:10 PM

Originally Posted by Macleef

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

$212 = 70 - Ae^{0}$

$A = {\color{red}212 - 70}$

$A = 142$

$15 = 70 - 142e^{-k2}$

$ln{\frac{\frac{(15-70)}{-142}}{-2}} = k$

$0.47 = k$

$f(4) = 70 - 142e^{-0.47\times4}$

$f(4) = 48.697F$

...is my answer correct?

49 degrees Fahrenheit seems a bit too cold for a cup of coffee!! XD

Your mistake is in red. It should be $A=70-212=-142$ . That would mean then that $f\left(t\right)=70+142e^{-kt}$

Try the problem from here. You will have to find a new k value before you can get to the desirable answer!

**Macleef** · January 24th 2009, 05:11 PM

haha that's what I thought too... thanks! I dislike making careless mistakes

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