Cost: Cell Phones. A cell phone company charges 50 for an...

Cost: Cell Phones. A cell phone company charges $\$ 50$ for an 800-minute monthly plan, plus an additional $\$ 0.22$ per minute for every minute over 800 . If a customer's bill ranged from a low of $\$ 67.16$ to a high of $\$ 96.86$ over a 6 -month period, what were the most minutes used in a single month? What were the least?

Key Concepts

Linear Functions

Linear functions describe relationships where one variable changes at a constant rate with respect to another. In cost models, a fixed fee combined with an extra charge per unit creates a straight-line relationship between usage beyond a threshold and the total cost incurred.

Fixed Costs

Fixed costs are amounts that remain constant regardless of usage or production levels. In a monthly cell phone plan, the base fee represents a fixed cost that the customer pays regardless of how many minutes are used.

Variable Costs

Variable costs change in direct proportion to the level of usage. In the cell phone billing scenario, the additional charge per minute over the allocated amount is a variable cost, meaning the total bill increases steadily as extra minutes are used.

Solving Linear Equations

Solving linear equations involves isolating unknown variables through algebraic operations. This process is used to determine the number of extra minutes used by equating the additional cost (total cost minus fixed fee) to the per-minute rate times the number of extra minutes.

Piecewise Functions

Piecewise functions are defined by different expressions over separate intervals. In billing problems like this, the cost function is piecewise: one part of the function covers the fixed fee for up to a certain number of minutes, and another part applies a linear rate for usage beyond that threshold.

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