Rental car company A charges a flat rate of 45 per day to...

Rental car company A charges a flat rate of $\$ 45$ per day to rent a car, with unlimited mileage. Company B charges $\$ 25$ per day plus $\$ 0.25$ per mile. (a) Find an expression for the cost of a car rental for one day from Company $\mathrm{A}$ as a linear function of the number of miles driven. (b) Find an expression for the cost of a car rental for one day from Company $\mathrm{B}$ as a linear function of the number of miles driven. (c) Determine algebraically how many miles must be driven so that Company A charges the same amount as Company B. What is the daily charge at this number of miles? (d) $\quad$ Confirm your algebraic result by checking it graphically.

Rental car company A charges a flat rate of $\$ 45$ per day to rent a car, with unlimited mileage. Company B charges $\$ 25$ per day plus $\$ 0.25$ per mile.
(a) Find an expression for the cost of a car rental for one day from Company $\mathrm{A}$ as a linear function of the number of miles driven.
(b) Find an expression for the cost of a car rental for one day from Company $\mathrm{B}$ as a linear function of the number of miles driven.
(c) Determine algebraically how many miles must be driven so that Company A charges the same amount as Company B. What is the daily charge at this number of miles?
(d) $\quad$ Confirm your algebraic result by checking it graphically.

Key Concepts

Graphical Verification of Intersection Points

Graphical methods involve plotting the linear functions on a coordinate plane to visually identify the point of intersection. This intersection represents the exact scenario where the costs are equal for both models, serving as a confirmation of the algebraic solution.

Solving Linear Equations to Determine Break-Even Points

When two different linear cost models are compared to find where they are equal, a system of linear equations must be solved. This process determines the break-even point—the point where both models incur the same total cost.

Slope-Intercept Form

The slope-intercept form of a linear equation, y = mx + b, is used to clearly define the rate of change (slope, m) and the initial value (y-intercept, b). This form is crucial in setting up cost functions where one parameter is the constant fee and the other scales with usage.

Fixed Cost and Variable Cost

Fixed cost refers to a baseline charge that does not change with usage (such as a daily rental fee), while variable cost is the portion that changes in direct proportion to the quantity used (like the cost per mile driven). Together, they form a linear function that models total cost.

Linear Functions

A linear function is a mathematical expression that describes a straight-line relationship between two variables, commonly written as y = mx + b. In cost modeling scenarios, it is used to represent situations where there is a constant rate of change, like a fixed cost plus a cost per unit of usage.

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