$*($Refer to the previous worksheet.$)$A kitchenware manufacturer has set up an LP problem to determine how many of type A $($Xa$)$ and type B $($X$)$ cups to produce to maximize the total profit. The LP formulation and the sensitivity report generated by Excel Solver are given on the right.a$)$ Determine the optimal solution and the maximized total profit.b$)$ Write the Excel formula set in the cell of the total profit $($suppose that the objective function coefficients, $$50$ and $$60,$ are stored in cells C$3$ and D$3$ respectively$).$c$)$ Without solving the LP problem again, find the optimal solution and the maximized total profit if both the unit profits of A and B become $$55.$d$)$Find the shadow price of the MOQ constraint $($K$).$e$)$Without solving the LP problem again, find the maximized total profit if the RHS of the Cutting, Assembly and MOQ constraints become $3$ hours, $5$ hours and $8$ respectively. Discuss whether the optimal solution remains unchanged.