$1.$ Apple aims to develop two VR headsets, Vision Pro and Vision Lite, for next year. These devices are manufactured in their plants in China and India. The Chinese facility's operating costs are $$90,000/$day$,$ and it can produce $200$ Vision Pros$/$day and $500$ Vision Lites$/$day$.$ At the Indian facility, the operating costs are $$80,000/$day$,$ and it can produce $50$ Vision Pros$/$day and $600$ Vision Lites$/$day$.$ Apple estimates it will sell $5,000$ Vision Pros and $40,000$ Vision Lites in $2025.$

$$ Formulate and solve LP model that minimizes the overall costs. How many days should each facility operate to fulfill the orders?

$$ Identify the binding constraints and slack.

$$ If a new manufacturing technique in China increases the number of Vision Lites produced daily to $600,$ how many days should each facility be operated?

$2.$ If an integer constraint is applied to the decision variable $($number of days$)$ in the above problem. How many days should each facility operate to fulfill the orders?

$3.$ For the original problem stated in Question $1,$ if the operating cost at the China facility is given by the equation $40000+($operating days$)2$ and the India facility is $15000+5*($operating days$)2.$ How many days should each facility operate to fulfill the orders? Identify the binding constraints and slack variables.

Please show steps for answers.