Concern the cost, C, of renting a car from a company which...

Concern the cost, $C,$ of renting a car from a company which charges $\$ 40$ a day and 15 cents a mile, so $C=f(d, m)=40 d+0.15 m,$ where $d$ is the number of days, and $m$ is the number of miles. (a) Find $f(3,200)$ and interpret it. (b) Explain the significance of $f(3, m)$ in terms of rental car costs. Graph this function, with $C$ as a function of $m$. (c) Explain the significance of $f(d, 100)$ in terms of rental car costs. Graph this function, with $C$ as a function of $d$.

Concern the cost, $C,$ of renting a car from a company which charges $\$ 40$ a day and 15 cents a mile, so $C=f(d, m)=40 d+0.15 m,$ where $d$ is the number of days, and $m$ is the number of miles.
(a) Find $f(3,200)$ and interpret it.
(b) Explain the significance of $f(3, m)$ in terms of rental car costs. Graph this function, with $C$ as a function of $m$.
(c) Explain the significance of $f(d, 100)$ in terms of rental car costs. Graph this function, with $C$ as a function of $d$.

Key Concepts

Cost Function Modeling

This concept involves representing real-world pricing structures using mathematical functions. In rental car cost problems, the total cost is modeled as a function of fixed fees (daily rental rate) and variable fees (mileage rate). This approach helps in understanding how changes in different cost components influence the overall price.

Function Evaluation

Function evaluation is the process of substituting given values into a function to obtain a specific result. In the context of rental car costs, it means calculating the expense by substituting specific values for the number of days and miles driven, which shows the real-time cost incurred by the renter.

Partial Evaluation of Multivariable Functions

This concept involves fixing one variable in a function of several variables to study the function’s behavior with respect to the other variable. For example, by setting the number of days or miles to a constant value, one can analyze how the cost changes solely with mileage or days, isolating the effect of one component.

Linear Functions and Graphing

Linear functions are mathematical expressions with a constant rate of change, represented graphically by straight lines. Graphing cost functions as linear equations allows one to visually interpret fixed and variable costs. In rental car cost problems, the graphs illustrate how the total cost increases as either the number of days or the number of miles increases.

Interpretation of Slope and Y-intercept

The slope and y-intercept are key components of a linear function, where the slope indicates the rate of change of the dependent variable with respect to the independent variable, and the y-intercept represents the initial value when the independent variable is zero. Understanding these parts provides insight into the marginal costs (e.g., cost per mile or per day) and the base charges involved in the pricing structure.