A basic cellphone plan costs dollar 20 per month for 60...

A basic cellphone plan costs dollar 20 per month for 60 calling minutes. Additional time costs dollar 0.40 per minute. The formula $$C=20+0.40(x-60)$$ gives the monthly cost for this plan, $C$, for $x$ calling minutes, where $x>60$. How many calling minutes are possible for a monthly cost of at least dollar 28 and at most dollar 40 ?

A basic cellphone plan costs dollar 20 per month for 60 calling minutes. Additional time costs dollar 0.40 per minute. The formula $$C=20+0.40(x-60)$$
gives the monthly cost for this plan, $C$, for $x$ calling minutes, where $x>60$. How many calling minutes are possible for a monthly cost of at least dollar 28 and at most dollar 40 ?

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Key Concepts

Domain Restrictions in Function Models

Many functions, especially those modeling real-world scenarios, are defined only for a certain range of the independent variable. For example, a cellphone plan that charges extra only after a certain number of minutes implies that the function is valid only when the usage exceeds that number. Recognizing these domain restrictions is important to ensure that solutions are realistic and applicable to the situation at hand.

Solving Inequalities

Solving inequalities involves finding the set of values that satisfy a condition expressed by an inequality. In cost and usage problems, inequalities are used to determine the range of usage values that result in a total cost falling between a specified minimum and maximum. The process includes isolating the variable and considering the correct direction of the inequality while performing algebraic manipulations.

Linear Cost Functions

A linear cost function is an equation that represents the total cost as a fixed fee plus a variable cost that is proportional to usage. In many real-world scenarios such as cellphone plans, one encounters a base fee for a set number of units (like minutes) and an additional cost for each unit beyond that allowance. This function typically has the form C = fixed cost + rate * (usage - free allowance).

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