Example $2.$ Zprint, a wireless phone service provider, offers three different plans for individuals. A user that chooses the Unlimited Plan pays $$99\mathrm{.}99$ flat fee per month to make unlimited calls. An individual who purchases Plan$-$A pays $$69\mathrm{.}99$ per month and gets $450$ minutes of service for free. Any calls beyond the first $450$ minutes are charged $$0\mathrm{.}45$ per minute. An individual who purchases Plan$-$B pays $$89\mathrm{.}99$ per month, and gets $900$ minutes of service for free. Any calls beyond the first $900$ minutes are charged $$0\mathrm{.}40$ per minute.

$($a$)$ A caller uses her wireless phone for $490$ minutes per month. Which plan should she choose to minimize the cost of her monthly wireless service?

$($b$)$ What is the minimum amount of service required $($in minutes$)$ by an individual for the Unlan to be the cheapest option?