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Advanced Math/Newton's Law of Cooling

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Expert: - 3/6/2010


Question
A pan of warm water (46 degrees Celsius) was put into a refrigerator. Ten minutes later, the water's temperature was 39 degrees. 10 minutes after that, it was 33 degrees Celsius. use Newton's Law of Cooling to estimate how cold the refrigerator was...

I tried solving it, and got two equations
39-T(surroundings)= (4t-T(surroundings))e^(-10t)
33-T(surroundings)= (4t-T(surroundings))e^(-20t)

but can't do the arithmetic right to solve it....Please help! Thank you in advance!

Answer
The equation is T(t) = (To)(e^(at)) fpr some constants To and a.

The value is given that when t=10, T(10) = 39.
The value is also given that when t=20, T(20) = 33.

The equations to solve are two, and the number of unknowns are 2.

Note tat To = 39e^(-10a) and that To = 33e^(-20a).
Since both of these equations are for To, the exponentials can be set equal to each other.
If the the ln() is taken of both sides, note the ln(cd) = ln(c) + ln(d).
In this way, ln(a) can be separated out and solved for.

Take the exponential of both sides and there is the value for a.
Put that back in to one of the equations near the top to find To.
PUt that value in the other equation to check and make sure that To was found correctly.  

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