production

A strawberry growing company is deciding its production and sale plan for the national and international markets.
The sale price for each ton of strawberry depends on the quantity offered in the market. If x1 tons is offered for the domestic market, the sale price will be (30 - x¹) CU / ton, while if x2 tons is offered for the international market, the sale price will be (40 - x²) CU / ton.
The cost for each ton of strawberry for the domestic market is 10 MUs, while for the international market it is 15 MUs.
The company has the capacity to produce up to 10 tons of strawberries for sale and, according to SAG restrictions, it must dedicate at least 10% of its production to the international market.
For technical production reasons, the company must additionally satisfy the following restriction: x₁² + x₂² ≤64.
a) Raise the NLP model that allows maximizing the net profit for the company
b) State the KKT conditions for the problem and indicate whether they are necessary and / or sufficient.
c) Analyze all the points that could satisfy the KKT conditions and determine the optimal solution of the problem. Describe graphically what happens in each case.
d) There is the option of buying new machinery to increase the production capacity of the company. In what range should the new machine increase production capacity to suit the company? How much would you be willing to pay for it?