Suppose that the monthly cost of maintaining a car is...

Suppose that the monthly cost of maintaining a car is linearly related to the number of miles driven during that month, and that for a particular model, on average, it costs 160 dollar per month to maintain a car driven 540 miles and 230 dollar per month to maintain a car driven 900 miles. (a) Write an equation relating the cost $c$ of monthly maintenance and the number $n$ of miles driven that month. (b) What would be the average cost of maintaining a car driven 750 miles per month? (c) On the basis of this relationship, if your average monthly maintenance is 250 dollar how many miles do you drive per month? (d) Sketch a graph of this equation using the horizontal axis for $n$ and the vertical axis for $c$ (e) What is the $c$ -intercept of this graph? How would you interpret the fact that even if no miles are driven there is still a monthly maintenance cost?

Suppose that the monthly cost of maintaining a car is linearly related to the number of miles driven during that month, and that for a particular model, on average, it costs 160 dollar per month to maintain a car driven 540 miles and 230 dollar per month to maintain a car driven 900 miles.
(a) Write an equation relating the cost $c$ of monthly maintenance and the number $n$ of miles driven that month.
(b) What would be the average cost of maintaining a car driven 750 miles per month?
(c) On the basis of this relationship, if your average monthly maintenance is 250 dollar how many miles do you drive per month?
(d) Sketch a graph of this equation using the horizontal axis for $n$ and the vertical axis for $c$
(e) What is the $c$ -intercept of this graph? How would you interpret the fact that even if no miles are driven there is still a monthly maintenance cost?

Key Concepts

Linear Functions

A linear function is a mathematical relationship between two variables where the change in one variable is directly proportional to the change in the other. It is typically expressed in the form y = mx + b, where m represents the slope and b represents the y-intercept. This concept is fundamental in algebra and is widely used to model real-world situations where relationships between quantities are assumed to be constant over the range of interest.

Slope (Rate of Change)

The slope of a linear function quantifies the rate at which the dependent variable changes with respect to the independent variable. It is calculated as the ratio of the change in the y-value to the change in the x-value between two points. In applications such as cost modeling or any scenario showing consistent increases or decreases, the slope reveals how much one quantity changes for each unit change in the other.

Y-Intercept (Constant Term)

The y-intercept in a linear function is the value of the dependent variable when the independent variable is zero. It represents the starting point or the fixed cost that exists independent of any additional influence represented by the independent variable. Interpreting the y-intercept aids in understanding inherent baseline values in various contexts, such as a fixed cost even when representing a scenario with zero activity.

Graphical Representation of Linear Equations

Graphing linear equations involves plotting points that satisfy the equation and drawing a straight line through these points. The horizontal axis typically represents the independent variable and the vertical axis represents the dependent variable. This visual representation provides a clear picture of the relationship between the variables, aiding in the interpretation of both the slope and the intercept, and making it easier to perform tasks like interpolation or extrapolation.

Interpolation and Extrapolation

Interpolation involves estimating values within the range of observed data points in a linear relationship, while extrapolation involves estimating values outside that range. These techniques are built on the assumption that the established linear trend continues, allowing predictions and extending the model to new conditions. They are essential tools for applying mathematical models to real-world scenarios where data may only be available at discrete intervals.

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