What is the temp of the thermometer at 1 min? How long to reach 15 degree F?
A thermometer is removed from a room where the temperature is 70 degrees F and is taken outside, where the air temperature is 10 degrees F. After one-half minute the thermometer reads 55 degrees F.
What is the reading of the thermometer at t = 1 min?
How long will it take for the thermometer to reach 15 degrees F?
Filed under: Cooking Thermometers | Jun 30th, 2009.
Use Newton's Law of Cooling:
T = T(environment) + (T(initial) – T(environment)) * e^(-r * t)
where
T = current temp
T(environment) = 10
T(initial) = 70
r = rate of cooling
t = time
You know that after t = .5 min, T = 55. Plug these values in and solve for r:
55 = 10 + (70 – 10) * e^(-.5r)
45 = (60) * e^(-.5r)
.75 = e^(-.5r) <- take the natural log of both sides
-.288 = -.5r
>>> r = .575
Now that you know r, solve for T at t=1min:
T = 10 + (70 – 10) * e^(-.575*1)
T = 10 + 60 * .563
>>> T = 43.8 degrees
Likewise solve for t when T=15
15 = 10 + (70 – 10) * e^(-.575t)
5 = 60e^(-.575t)
.0833 = e^(-.575t) <- take the natural log of both sides
-2.49 = -.575t
>>> t = 4.32 min