The following data is available from the batch fermentation process,

where ethanol is produced by S$.$ cerevisiae in the bioreactor. Data was collected at $6$ different time points.

Time $($hour$)$ : $(2,4,6,8,10,12),$ Ethanol $($g$/$L$)$: $(1,4,10,18,25,27)$

You want to construct two different models based on the data to predict ethanol concentration as a

function of time: a linear model $(=$ a$0+$ a$1$x$)$ and a cubic model $(=$ a$0+$ a$1$x$^2+$ a$2$x$^3).$

$[$Solve in computer$]$ Use a programming language $($python or R$)$ to solve equation $(2)$ in computer as follows:

a$)$ Calculate the parameters of the linear model from the matrix equation derived in the model $($use the already calculated values for the elements in the equation$).$ Calculate also R$2$ using the formula given in the class.

b$)$ Calculate the values of the coefficient matrix in computer for the cubic model using the formula you derived in $2-$b$.$ Later, calculate the parameters and R$2$ for the cubic model