a piece of metal is heated to 300 degrees and is put into a room held at constan

Question

a piece of metal is heated to 300 degrees and is put into a room held at constant temperature of 0 degreees at time t=0. after one minute the bar has a temperature of 220 degrees. How much MORE time must the bar be kept in the 0 degree room to bring its temperature down to 100 degrees?

a. use the initial temperature, room temperature, and temperature after 1 min to determine the temperature function T of the bar.

b. Solve T(t) = 100 for t to determine how many minutes the bar has to be in the 0 degree room to reach a temperature of 100 degrees.

c. Now, answer the question above.

Explanation / Answer

As newtons law of cooling is given by

T -Ta= (To-Ta)exp(-at)

Now, at t = 1 min, To = 300 comes down to T = 220

And Ta = 0

Hence,

220 = 300exp(-a)

Or

ln(220/300) = -a

a = ln(300/220) = 0.31

Now, we require the time that metal takes place to come down to 100 degeers from 220

Hence, now To = 220, T = 100, Ta = 0

a= 0.31

Using newtons law of cooling,

100 = 220exp(-0.31 x t)

Or

ln(100/220) = -0.31 t

Hence,

t = ln (220/100)/0.31

t = 2.54 minutes

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