The price for a round-trip bus ride from a university to...

The price for a round-trip bus ride from a university to center city is 2.00 dollars, but it is possible to purchase a monthly commuter pass for 25.00 dollars with which each round-trip ride costs an additional 0.25 dollars. (a) Find equations for the cost $C$ of making $x$ round-trips per month under both payment plans, and graph the equations for $0 \leq x \leq 30$ (treating $C$ as a continuous function of $x$, even though $x$ assumes only integer values). (b) How many round-trips per month would a student have to make for the commuter pass to be worthwhile?

The price for a round-trip bus ride from a university to center city is 2.00 dollars, but it is possible to purchase a monthly commuter pass for 25.00 dollars with which each round-trip ride costs an additional 0.25 dollars.
(a) Find equations for the cost $C$ of making $x$ round-trips per month under both payment plans, and graph the equations for $0 \leq x \leq 30$ (treating $C$ as a continuous function of $x$, even though $x$ assumes only integer values).
(b) How many round-trips per month would a student have to make for the commuter pass to be worthwhile?

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Key Concepts

Linear Functions

Linear functions represent relationships where one variable changes at a constant rate with respect to another. In cost analysis, they are used to model situations where there is a fixed cost component along with a variable cost per unit, making it straightforward to predict total cost for any quantity of usage.

Graphing Linear Functions

Graphing linear functions allows the visualization of cost relationships by plotting the total cost against the number of units (such as trips) taken. This visual representation helps in comparing different scenarios or pricing plans and understanding how changes in usage levels affect overall costs.

Break-even Analysis

Break-even analysis involves determining the point at which two different pricing or cost structures yield the same total cost. In such problems, setting the two cost functions equal to each other and solving for the variable identifies the threshold where one option becomes more cost-effective than the other.

Systems of Equations

Systems of equations are used to solve for the intersection point of multiple linear functions. By equating two cost equations and solving for the number of units, one can find the exact point at which the expenses of two different payment plans are equal, which is crucial in decision-making scenarios.

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