Howard J. answered • 10/14/19
Principal Mechanical Engineer with >30 years' math coaching experience
According to , the rate of change of an object's temperature is proportional to the difference between the temperature of the the object and that of the surrounding medium. The accompanying figure shows the graph of the temperature (in degrees Fahrenheit) versus time (in minutes) for a cup of coffee, with initial temperature 200 degrees Fahrenheit, that is allowed to cool in a room with a constant temperature of 75 degrees Fahrenheit.
I'm going blind without the figure.
The problem says dT/dt = B(T-Ta) where T is the temperature of the object in °F, B is a constant, t is time in minutes, and Ta=the temperature of the surrounding fluid (air) = 75°F. We are given T(0) = 200 °F.
We can estimate this with:
ΔT/Δt=B(T-Ta)
(a) Estimate T when t=10 minutes: ____
ΔT/Δt≅dT/dt=B(T-Ta)
[T(10)-T(0)]/10=B[T(10)-75]
[T(10)-200]=10B[T(10)-75]
Since we don't know B, we can't calculate T(10). I suppose the figure would have helped.
(b) Estimate dT/dt when t=10 minutes: ____
Newton's Law of Cooling can be expressed as dT/dt = k(T-T0) , where k is the constant of proportionality and T0 is the temperature of the surrounding medium.
If we had T(10) from part A, then dT/dt≅ΔT/Δt=[T(10)-200]/10
(c) Use the results of parts (a) and (b) to estimate the value of k:
value of k:_____
From part (a) we would have found T(10). From part (b) we have ΔT/Δt.
Using these,
k= (ΔT/Δt)/[T(10)-200]
