Uncertainty in Newton's law of cooling

[sunmoonlight]

Homework Statement

Uncertainty in Newton's law of cooling

Relevant Equations

T(t) = = 𝑇_𝐴+(𝑇_𝑜−𝑇_𝐴)𝑒^(−𝑘𝑡)

I'm finding the uncertainty of k, given that each temperature has an uncertainty of +/- 0.5 degress.

 

[haruspex]

Homework Statement: Uncertainty in Newton's law of cooling
Relevant Equations: T(t) = = 𝑇_𝐴+(𝑇_𝑜−𝑇_𝐴)𝑒^(−𝑘𝑡)

I'm finding the uncertainty of k, given that each temperature has an uncertainty of +/- 0.5 degress.

You will also need approximate values for the temperatures.
Per forum rules, please show some attempt.

 

[sunmoonlight]

say the T(O) = 90 +/- 0.5, T(t): 60 +/- 0.5, TA = 10 +/- 0.5, temp difference (T(t) - TA) is 50 degrees +/- 0.5, t= 100s
1. Is the uncertainty for ln (T(t) - TA) = 1/2*(ln50.5 - ln49.5) = +/-0.01?
2. If you substitute the values into the eqt, you get k = (ln50/80)/-100, so what's the uncertainty for k (like how do you find uncertainty involving logs?)

 

[haruspex]

say the T(O) = 90 +/- 0.5, T(t): 60 +/- 0.5, TA = 10 +/- 0.5, temp difference (T(t) - TA) is 50 degrees +/- 0.5, t= 100s
1. Is the uncertainty for ln (T(t) - TA) = 1/2*(ln50.5 - ln49.5) = +/-0.01?
2. If you substitute the values into the eqt, you get k = (ln50/80)/-100, so what's the uncertainty for k (like how do you find uncertainty involving logs?)

There are different concepts of uncertainty. An engineer worried about engineering tolerances would just look at the combinations of the extreme values. A scientist would take the given uncertainties as standard deviations in normal distributions and use root-sum-square approaches to combine them. I assume you are looking for the latter.

Can you find the uncertainty in $e^{-kt}$?