a) The organizing commitee of a cricket match wants to maximize the profit P from the sales of the tickets sold. The cricket stadium will have three types of seating arrangement with their corresponding ticket rates. The maximum seating capacity of the stadium is 15000. The committee wants to allocate the number of seats in the stadium for each category so that profit P is maximum which is defined as P=500A^{0.45}B^{0.4}C^{0.2} where A, B and C are the number of seats in each category. It is decided that the ticket rates in category A, B and Care Rs. 550, Rs. 1800 and Rs. 3000 respectively.

i) What are the different constraints that needs to be applied to solve the problem?

2) Find the optimum number of seats in each category by optimizing the profit subject to above constraints, and the maximum profit.