After the included minutes have been exhausted, a cell phone...

After the included minutes have been exhausted, a cell phone company charges an additional $\$ 0.08$ per minute. Plan A uses a flat rate of $\$ 0.10$ per minute. Which plan is the least costly if a person uses 750 minutes per month? $$\begin{array}{|c|c|c|} \hline \text { Plan } & \begin{array}{c} \text { Monthly } \\ \text { Fee } \end{array} & \begin{array}{c} \text { Induded } \\ \text { Minutes } \end{array} \\ \hline \mathrm{A} & \$ 0 & \text { None } \\ \hline \mathrm{B} & \$ 29.99 & 500 \\ \hline \mathrm{C} & \$ 39.99 & 1,000 \\ \hline \mathrm{D} & \$ 49.99 & 1,500 \\ \hline \end{array}$$ A Plan A C Plan C B Plan B D Plan D

After the included minutes have been exhausted, a cell phone company charges an additional $\$ 0.08$ per minute. Plan A uses a flat rate of $\$ 0.10$ per minute. Which plan is the least costly if a person uses 750 minutes per month?
$$\begin{array}{|c|c|c|}
\hline \text { Plan } & \begin{array}{c}
\text { Monthly } \\
\text { Fee }
\end{array} & \begin{array}{c}
\text { Induded } \\
\text { Minutes }
\end{array} \\
\hline \mathrm{A} & \$ 0 & \text { None } \\
\hline \mathrm{B} & \$ 29.99 & 500 \\
\hline \mathrm{C} & \$ 39.99 & 1,000 \\
\hline \mathrm{D} & \$ 49.99 & 1,500 \\
\hline
\end{array}$$
A Plan A
C Plan C
B Plan B
D Plan D

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Key Concepts

Break-even Analysis

Break-even analysis involves finding the point at which two different pricing plans cost the same amount. This technique is useful when deciding which plan offers the better deal based on expected usage levels.

Linear Equations

Since many pricing plans use a constant rate per unit after a certain threshold, the total cost can be modeled using linear equations. This allows for straightforward calculations and comparisons between different pricing structures.

Cost Functions

Cost functions represent the relationship between the number of units used and the total cost incurred. They combine both fixed charges (such as a monthly fee) and variable charges (such as a per-minute fee) into a mathematical expression, which helps in comparing different pricing plans based on usage.

Fixed Costs vs. Variable Costs

Fixed costs are charges that remain constant no matter how many units are consumed, while variable costs change in direct proportion with usage. Understanding this distinction is crucial when evaluating plans that mix a monthly fee with per-unit charges.

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