A discount pass for a bridge costs 21 per month. The toll...

A discount pass for a bridge costs $\$ 21$ per month. The toll for the bridge is normally $\$ 2.50$, but it is reduced to $\$ 1$ for people who have purchased the discount pass. a. Express the total monthly cost to use the bridge without a discount pass, $f,$ as a function of the number of times in a month the bridge is crossed, $x$ b. Express the total monthly cost to use the bridge with a discount pass, $g,$ as a function of the number of times in a month the bridge is crossed, $x$. c. Determine the number of times in a month the bridge must be crossed so that the total monthly cost without the discount pass is the same as the total monthly cost with the discount pass. What will be the monthly cost for each option?

A discount pass for a bridge costs $\$ 21$ per month. The toll for the bridge is normally $\$ 2.50$, but it is reduced to $\$ 1$ for people who have purchased the discount pass.
a. Express the total monthly cost to use the bridge without a discount pass, $f,$ as a function of the number of times in a month the bridge is crossed, $x$
b. Express the total monthly cost to use the bridge with a discount pass, $g,$ as a function of the number of times in a month the bridge is crossed, $x$.
c. Determine the number of times in a month the bridge must be crossed so that the total monthly cost without the discount pass is the same as the total monthly cost with the discount pass. What will be the monthly cost for each option?

Key Concepts

Linear Functions

Linear functions are mathematical expressions that model relationships where the change in one variable is directly proportional to the change in another. They are often written in the form y = mx + b, where m represents the slope (rate of change) and b represents the y-intercept (the initial value when the independent variable is zero). In cost modeling, this format succinctly represents how costs vary with usage.

Cost Modeling

Cost modeling involves creating functions to represent the total cost associated with different scenarios, typically separating fixed costs and variable costs. In this context, one model includes only a variable cost per use, while another includes a fixed monthly fee plus a lower variable cost per use. This distinction allows for a clear comparison between different payment strategies based on usage levels.

Break-even Analysis

Break-even analysis is the process of finding the point at which two different cost models result in the same total cost. It involves setting the two cost functions equal to each other and solving for the independent variable, which typically represents the quantity of usage. The break-even point indicates when the investment in a cost-saving alternative, such as a discounted pass, becomes financially beneficial compared to the alternative.

Solving Linear Equations

Solving linear equations is an essential algebraic technique used to find the value of the unknown variable that satisfies the equation. In the context of cost comparison, equating two linear cost functions results in a simple linear equation that can be manipulated by isolating the variable. This step is critical to determining the exact usage level where both cost strategies yield the same expense.

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