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## Homework Statement

A small metal bar, whose initial temperature was 20 degrees C is dropped into a large container of boiling water. How long will it take the bar to reach 90 degrees C if it is known that its temperature increases 2 degrees in 1 second. How long will it take the bar to reach 98 degrees C?

(the back of the book gives 82.1 seconds and 145.7 seconds)

## Homework Equations

[tex]\frac{dT}{dt}=k(T-Tm)[/tex]

where [tex]T_m[/tex] is the temperature of the surroundings.

## The Attempt at a Solution

[tex]\int{\frac{1}{T-T_m}\frac{dT}{dt}dt=\int kdt[/tex]

[tex]T(t)=Ce^{kt}+T_m[/tex]

I'm given the points [tex]T(0)=20[/tex] and [tex]T(1)=T(0)+2=22[/tex]

The problem is that this give two equations and

__three__unknowns:

[tex]C+T_m=20[/tex]

[tex]Ce^k+T_m=22[/tex]

I'm given that the water is boiling, but that only gives a minimum temperature of 100 degrees celsius.