$[$Solve by hand$-$writing$]$ 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.

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

function of time: a linear model $((hat(y))=a_0+a_{1}x)$ and a cubic model $((hat(y))=a_0+a_1{x}^2+a_2{x}^3).$

a$)$ Use the matrix equation derived in the class to predict the parameters for the linear model using

Gaussian elimination. Calculate also $R^2$ using the formula given in the class.

b$)$ Start from the objective function that defines the error. Derive the matrix equation for the cubic

model. Show all the steps. Predict the parameter values using Gauss Jordan elimination. Calculate $R^2$

value, and compare the goodness of the model with the linear model you used in $($a$).$

show all row echelon form and solve all the subquestions!