Answer to Question #273943 in Differential Equations for Sanzu

Question

A metal bar at a temperature of 100 deg. fahrenheit is placed in a room at a constant temperature of 0 deg. fahrenheit. If after 20 minutes the temperature of the bar is 50 deg. Fahrenheit, find: a) the time it will take the bar to reach a temperature of 25 deg. Fahrenheit? b) the temperature of the bar after 10 minutes?

Answer 1

a) the time it will take the bar to reach a temperature of 25 deg. Fahrenheit?

We can use Newton's Law of Cooling: The temperature T of an object with initial temperature $T_0$ after $t$ minutes in a room with ambient temperature $T_r$   is given by:

$T=Ce^{-kt}+T_r$

Here  $T_r=0$ . We are given points $(t,T)$  as $(0,100)$ and $(20,50)$

$100=Ce^{0\times t}\implies C=100$

$50=100e^{-20k} \implies k={ln(0.5)\over -20}\approx 0.035$

$\implies T=100e^{-0.035t}$

To get the time it will take the bar to reach a temperature of 25 deg. Fahrenheit

$T=100e^{-0.035t}$

$25=100e^{-0.035t}$

$e^{-0.035t}={25\over 100}$

$ln(e^{-0.035t})=ln({25\over 100})$

$-0.035t=ln({25\over 100})\implies t={ln({25\over 100})\over -0.035}=39.6$ minutes

b) To get the temperature of the bar after 10 minutes

$T=100e^{-0.035\times 10}=70.5\degree F$