Use the information from Exercise 25 to answer the following...

Use the information from Exercise 25 to answer the following questions.
a. If Budget doubled the cost per mile, how would that change the equation found in Exercise 25 part a?
b. Would the cost to drive 75 miles also double? Explain your reasoning.

Key Concepts

Linear Functions

Linear functions are mathematical models represented by equations of the form y = mx + b, where m is the slope and b is the y-intercept. In cost problems, the slope represents the cost per unit (such as cost per mile) while the y-intercept represents a fixed cost that is incurred regardless of usage.

Slope as a Rate

In the context of the problem, the slope of the linear equation corresponds to the cost per mile. Doubling the cost per mile means that the slope in the equation will be doubled. This change directly modifies how the variable cost accumulates with each mile driven, altering the relationship between distance and total cost.

Fixed Costs

Fixed costs are the components of a cost function that do not change with the level of usage or activity (such as the base fee in a taxi fare scenario). When only the cost per mile (slope) is altered, the fixed cost remains the same, meaning that while variable costs increase, the initial cost does not, which results in a total cost that may not exactly double for a given mileage.

Scaling and Proportionality

Scaling in a linear model involves understanding the effect of multiplying one of the terms by a constant. Doubling the cost per mile scales the variable component of the function by a factor of two; however, if there is a non-zero fixed cost, the overall total cost for a specific distance, such as 75 miles, will not double due to this unaltered constant term.

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