Gatson manufacturing company is willing to promote $2$ types of tires. Econory tire. Premium tire. and these two tires are independent of each other in terms of demand, cost. price, etc. in analytics team of this company has estimated the profit functions for both the tires as$.$

Monthly profit for Economy tire $=49\mathrm{.}2415$ LNIX;$)+180\mathrm{.}414$

Monthly profit for Premium tire $=84\mathrm{.}344$ LNe$)-150\mathrm{.}112$

where Xy and Xg are the advertising amount allocated to Economy tire and Premiur tire, respectively, and LN is the natural logarithm function. The advertising budget is $$200,000,$ and management has dictated that at least $$20\mathrm{.}000$ must be allocated to each of the two tires.

$($NOTE: To compute a natural logarithm for the value X in Excel, use the formula $=$ LNIX$).$ For Solver

to find an answer, you also need to start with decision variable values greater than $0$ in this problem.$)$

a$.$ Create a spreadsheet model that will prescribe how the company should allocate its marketing budget.

b$.$ Find the optimal solution in part a$.$

c$.$ Find the total profit.