Major software companies, after years of providing unlimited free telephone technical support for their products, have recently begun to charge for these services $($typically after an initial start$-$up period of $90$ days$).$  Most companies offer two pricing plans. For instance, Lotus Development offers users of their spreadsheet software the option of paying either $($i$)$ $$2\mathrm{.}00$ per minute for telephone support or $($ii$)$ a $$129$ flat charge for a year of unlimited toll$-$free calls.

a$.$ Consider a customer with a yearly demand for service support of $,$ where P is the price per minute and Q is the number of minutes of calls made per year. How many calls would this customer make under plan $($i$)?$  Why?

b$.$ How many calls would he make under plan $($ii$)?$  What would be the annual cost to this customer under each plan? Which of the two plans will he choose? Which customer will choose the plan that will bring him the largest consumer surplus? What is the amount of consumer surplus of these tariff plans?