Solve the following in python:

Problem $6$: Using a Model that Breaks

Newton's Law of Cooling gives the rate of a cooling of an object proportional to its current

temperature and the ambient temperature. The model described is:

Temp$(t)=(T_0-T_a)e^{-kt}+Ta$

where $T_0$ is the object's initial temperature, $T_s$ is the ambient $($or background$)$ temperature, $t$ is

time elapsed, the $k$ and experimentally determined constant. Assume $k=0\mathrm{.}19617.$ Implement

a function determine$\_$t $(T,TO,Ta,k)$ that, when given a final temperature $T,$ determines the time it

takes to reach that temperature. For example, this code

which has the initial temperature at $200F,$ ambient temperature at $72F,$ and final temperature

at $85F$ produces

which means it will take about $11$min$\mathrm{.}40sec.$ to cool. The problem with this model is that we

can't use it directly to ask how long will it take to cool to room temperature:

$T_a=(T_0-T_a)e^{-kt}+T_a$

$e^{-kt}=0$

We can't take the log of $0.$ We confirm with Python:

The reason is that this is a limit $($convergence$)$:

$\underset{t}{lim}{e}^{-kt}=0$

Indeed, if we run our solution we'll raise the same error:

How can we solve this problem? We can form the linear regression over, say, the last three

points:

The code uses the last three whole temperatures $($we$\text{'}$ll need our original function to deter$-$

mine them$).$ We can then use numpy's polyfit bttps:$//$aunpy$.$org$/$doc$/$stable$/$reference$/$

geverated$/$auny $.$polyfit. html that has how to tind the coefticients $y=mx+b$ as an array

and how to automatically build the function from the coefticients.

$x=0\mathrm{.}19617$

Assignment $N29$ Convergence, $Al($Clustering$),$ Monte Carlo

Implement the two functions.

Round to two decimal places.

Here is the defining for the problem:

#Problem $6$

#INPUT temperature, initial temp, ambient temp, k

#OUTPUT time to reach T from T$0$ in Ta

def determine$\_$t$($T$,$T$0,$Ta$,$k$)$:

#INPUT T$0,$ Ta$,$ k

#OUTPUT using the function above model the three data points

#using numpy polyfit and poly$1$d

def final$\_$temp$($T$0,$Ta$,$k$)$: