Consider a hospitality and leisure company operating a ski resort that employs distinct pricing strategies for peak (holidays and weekends) and off-peak (weekdays) visits. The demand function during peak time is denoted as 1 = 550 1/2 1, while for off-peak time, it is 2 = 208 1/5 2. The cost associated with running the ski resort is expressed as (1 + 2 ) 2 . Answer the following questions.

a) Solve the firm's profit maximisation problem and find out the maximum profit.

b) Considering the potential environmental impact of a ski resort, the local authorities decide to impose a tax on the overall number of visitors, represented as T= (1 + 2 ), where is a parameter and T is the total tax revenue generated from this ski business. Given the tax, the company still maximises its profit and the local authorities maximise their tax revenue from this ski resort. What is the maximum profit the company can achieve under these conditions? And what is the maximum tax revenue the local authorities can achieve under these conditions?

c) Rather than imposing a tax linked to the total number of visitors, the local authorities decree that the company bears the cost for their ski resort capacity (K) at a rate of 20 per unit of capacity. The constraints dictate that neither peak nor off-peak number of visitors can surpass this capacity. Using KKT conditions, explore all conceivable scenarios and determine the optimal number of capacity units the company should pay for, along with the corresponding charges for peak and off-peak time visitors. What is the maximum profit the company can achieve under these conditions?