Newton's law of cooling
T(t)= Tm + (To-Tm)e^-kt, k >0\
Evaluate T(0). Simplify and explain the result
I'm very confused on how to do this, i think i will be able to explain the result but if you could help me simplify this that would be great, Thank you very much.
Tm is the final temp after a long time as you can see from the equation. This is normally the temperature in the surrounding environment.
To is the starting temperature.
T(0) = Tm + To e^0 - Tm e^0
but e^0 is 1
so
T(0) = Tm + To - Tm
or
T(0) = To
or in other words the temp at T = 0 is the starting temp To
what else is new?
To evaluate T(0) in the equation T(t) = Tm + (To - Tm)e^(-kt), we need to substitute t = 0 into the equation. Let's do that step-by-step:
1. Start with the original equation:
T(t) = Tm + (To - Tm)e^(-kt)
2. Replace t with 0:
T(0) = Tm + (To - Tm)e^(-k*0)
3. Simplify the term e^(-k*0):
Since any number raised to the power of 0 is equal to 1, we have:
T(0) = Tm + (To - Tm) * 1
4. Simplify further:
T(0) = Tm + (To - Tm)
T(0) = Tm - Tm + To
T(0) = To
So, after simplification, we can conclude that T(0) = To. This means that at time t = 0, the temperature is equal to the initial temperature given by To.