Please formulate a linear programming model for this problem, i$.$ e$.$ include definition of the decision variables and their complete notation, then please provide the exact mathematical expressions, inequalities, equalities, etc. for the constraints and the objective function $($please state whether it is a maximization or minimization$).$ You may use the sigma $($summation$)$ notation and other notations to provide a compact presentation of the model.

The law firm of Smith and Jones is recruiting at law schools for new lawyers for the coming year. The firm has developed the following estimate of the number of hours of casework it will need its new lawyers to handle each month for the following year:

$\backslash$table$[[$Month$,$Casework $($hr$.),$Month,Casework $($hr$.)],[$January$,650,$July,$750],[$February$,450,$August,$900],[$March$,600,$September,$800],[$April$,500,$October,$650],[$May$,700,$Nowember,$700],[$June$,650,$December,$500]]$

Each new lawyer the firm hires is expected to handle $150$ hours per month of casework and to work all year. All casework must be completed by the end of the year. The firm wants to know how many new lawyers it should hire for the year.