Chemical firm David Inc produces two products: A and B$.$ Each ton of A is a mixture of $40\%$ of Material $1$ and $60\%$ of Material $3.$ Each ton of B is a mixture of $50\%$ of Material $1,15\%$ of Material $2,$ and the remaining $35\%$ of Material $3.$ After deducting relevant costs, every ton of product A generates a gross profit of $$35$ and every ton of product $B$ generates a gross profit of $$25.$ The firm's production is constrained by a limited availability of the three materials; Material $1$ amount available is $17$ tons, Material $2$ amount available is $2$ tons, and Material $3$ amount available is $15$ tons. The firm has a sales contract for $14$ tons of product A that has to be satisfied. Formulate the mathematical model that will maximize the firm's gross profit.