A company rents cars at 40 a day and 15 cents a mile. Its...

A company rents cars at $\$ 40$ a day and 15 cents a mile. Its competitor's cars are $\$ 50$ a day and 10 cents a mile. (a) For each company, give a formula for the cost of renting a car for a day as a function of the distance traveled. (b) On the same axes, graph both functions. (c) How should you decide which company is cheaper?

A company rents cars at $\$ 40$ a day and 15 cents a mile. Its competitor's cars are $\$ 50$ a day and 10 cents a mile.
(a) For each company, give a formula for the cost of renting a car for a day as a function of the distance traveled.
(b) On the same axes, graph both functions.
(c) How should you decide which company is cheaper?

Key Concepts

Break-Even Analysis

Break-even analysis is a technique used to determine the point at which two different cost schemes result in the same overall cost, meaning neither option is more expensive. This is done by setting two cost functions equal to each other and solving for the variable, indicating the usage level at which switching or comparing alternatives becomes beneficial.

Graphical Comparison of Functions

Graphing functions provides a visual method to compare different cost models. By plotting the cost functions on the same axis, one can directly observe the fixed cost differences and variable cost rates, as well as the break-even point where the costs are equal. This technique aids in decision-making by illustrating at what point one option becomes more cost-effective than the other.

Linear Functions

Linear functions are mathematical representations of relationships where the rate of change is constant. They are typically written in the form y = mx + b, where m is the slope and b is the y-intercept. In cost analysis, linear functions capture scenarios where there is a fixed fee plus a variable cost based on usage, providing a simple model for comparison.

Fixed and Variable Costs

Fixed and variable costs are components of cost functions that distinguish between expenses that remain constant regardless of usage (fixed costs) and those that change based on the level of activity (variable costs). This concept is essential when modeling rental costs or production costs, as it breaks down the total cost into a part that must be paid regardless of consumption and another that scales with consumption.

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