Throgmorton Accountancy Services specialise in two types of accountancy work; audit contracts and tax contracts.

Each audit contract requires $30$ hours of work from qualified accountants and $12$hours of work from junior accountants. The revenue per audit contract is $2,500$ and direct expenses relating to each contract amount to $600$ per contract.

Each tax contract requires $18$ hours of work from qualified accountants and $22$ hours of work from junior accountants. The revenue per tax contract is $2,250$ and direct expenses relating to each tax contract amount to $500$ per contract.

Accountants that work for the firm are employed on an hourly basis. Qualified accountants earn $45$ per hour and junior accountants earn $21$ per hour. Administration costs are attributed to contracts using a rate per contract hour and this totals $296$ for each audit contract and $137$ for each tax contract. The total administration costs are $80,000$ for a four week period.

During the four$-$week period ending on July $31$st $2023$ the number of working hours available are $2,000$ for qualified accountants and $1,650$ for junior accountants. Agreements are already in place for $20$ audit contracts with returning clients. Throgmorton Accountancy Services must start and complete these projects in the four$-$week period under consideration. A maximum of $60$ tax contracts may be undertaken during the four$-$week period to $31$st July $2023.$

REQUIRED:

a$.$ Calculate the contribution for each of the two types of contract.

b$.$ Determine the optimal mix of contracts for the four$-$week period to $31$st July $2023$ assuming that Throgmorton Accountancy Services are seeking to maximise profit. You should use a linear programming graph, identifying the feasible region and the optimal point using whatever equations you need.