$5.(40$ Points$)$ You have been recording the concentration of a contaminant in $\backslash (\backslash$mathrm$\{$mg$\}/\backslash$mathrm$\{$L$\}\backslash )$ as it travels down a stream over the course of $10$ days. You know that the contamination plume should behave according to the following equation:

$\backslash [$

C$=$C$\_\{0\}$ e$^\{-$C$\_\{1\}\backslash$left$($t$-$t$\_\{0\}\backslash$right$)^\{2\}\}$

$\backslash ]$

Where $\backslash ($ C$\_\{0\}\backslash )$ is the initial concentration, $\backslash ($ C$\_\{1\}\backslash )$ is the decay constant of the contaminant, and $\backslash ($ t$\_\{0\}\backslash )$ is the time at which the peak of concentration occurs. For this problem you know that the initial concentration was $\backslash (100\backslash$mathrm$\{$mg$\}/\backslash$underline$\{\backslash$underline$\{$L$\}\}\backslash )$ and the peak of the concentration occurred at $5$ days. The daily observations have been provided to you in a text file format $($e$.$g$.$ np$.$ loadt xt $(\text{'}$YourFileName$.$txt$\text{'})).$ For this problem you will need to figure out the decay constant $\backslash (\backslash$left$(\backslash$mathrm$\{$C$\}\_\{1\}\backslash$right$)\backslash )$ by trying $10$ different values staring from a value of $0$ all the way up to a value of $0\mathrm{.}9($i$.$e$.$ np$.$arange $(\backslash (0,1,0\mathrm{.}1\backslash ))).$ You will select the best one based on the goodness of fit metrics.

a$.(20$ Points$)$ To visualize the fit of the data you will need to calculate the four metrics for the fit $($e$.$g$.$ L$1,$ L$2,$ RMS$,$ and R$2).$ You will need to print these metrics to the console following the format provided below. You will need to plot each of the metrics on their own subplot in a single figure $($e$.$g$.$

fig. add subplot $(22$X$)).$ For full credit your plot must have the following elements:

i$.$ All four plots must be on one figure$-$that is one figure with for subplots defined by $2$ rows and $2$ columns

ii$.$ Each subplot must have the data plotted with black $+$ signs $($marker$=\text{'}+\text{'},$ color$=\text{'}$k$\text{'}.$s$=50).$

iii. Each subplot must have both axes clearly labeled

iv$.$ Each subplot must have the grid turned on

v$.$ I need to be able to read the labels. So either re$-$size the figure before saving or use fig.tight layout $().$