# Exponential Problem

• Jan 24th 2009, 05:05 PM
Macleef
Exponential Problem
Instant coffee is made by adding boiling water ($\displaystyle 212F$) to coffee mix. If the air temperature is $\displaystyle 70F$, Newton's law of cooling says that after $\displaystyle t$ minutes, the temperature of the coffee will be given by a function of the form $\displaystyle f(t) = 70 - Ae^{-kt}$. After cooling for 2 minutes, the coffee is still $\displaystyle 15F$ too hot to drink, but 2 minutes later it is just right. What is this ideal temperature for drinking?

$\displaystyle 212 = 70 - Ae^{0}$

$\displaystyle A = 212 - 70$

$\displaystyle A = 142$

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

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

$\displaystyle 0.47 = k$

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

$\displaystyle f(4) = 48.697F$

• Jan 24th 2009, 05:10 PM
Chris L T521
Quote:

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

$\displaystyle 212 = 70 - Ae^{0}$

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

$\displaystyle A = 142$

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

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

$\displaystyle 0.47 = k$

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

$\displaystyle f(4) = 48.697F$

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