Question

A cold metal bar at -30 degrees Celsius is submerged in a pool maintained at a temperature of 40 degrees Celsius. Half a minute later, the temperature of the bar is 20 degrees Celsius. How long will it take the bar to attain a temperature of 30 degrees Celsius?

I know that dy/dt=k(y-b) and y(t)=b+Ce^kt

Answer 1

The solution will be of the form

T - 40 = (-70)*exp(-kt)

Note that this satisfies T = -30 when t = 0 and T-> 40 as t -> infinity

Solve for k by using the fact that
T = 20 at t = 30 s.
-20 = -70*exp(-30 t)
exp(-30k) = 0.2857
-30k = -1.253
k = 4.18*10-2
(1/k) = 23.9 seconds

Now solve for t when T = 30 C

Answer 2

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