The function y = 3.75 + 1.5(x-1) can be used to determine the cost in dollars for a taxi ride of x miles. what is the rate of change of the cost in dollars with respect to the number of miles?
The Correct Answer and Explanation is:
Answer
The rate of change of the cost in dollars with respect to the number of miles is $1.50 per mile.
Explanation
In mathematics, the “rate of change” of a linear function describes how one quantity changes in relation to another. For a function that models cost versus distance, the rate of change tells us how much the cost increases for every one-unit increase in distance. In the context of a linear equation, the rate of change is represented by the slope of the line.
The given function is:
y=3.75+1.5(x−1)y=3.75+1.5(x−1)Here, y represents the total cost in dollars, and x represents the number of miles traveled. To determine the rate of change, we need to find the slope of this linear function. The most common form for a linear equation is the slope-intercept form, y = mx + b, where m is the slope (the rate of change) and b is the y-intercept.
We can convert the given equation into this form through algebraic manipulation:
- Start with the given function:
y=3.75+1.5(x−1)y=3.75+1.5(x−1) - Distribute the 1.5 across the terms inside the parentheses (x – 1):
y=3.75+1.5x−1.5(1)y=3.75+1.5x−1.5(1)y=3.75+1.5x−1.5y=3.75+1.5x−1.5 - Combine the constant terms (3.75 and -1.5) to simplify the equation:
y=1.5x+(3.75−1.5)y=1.5x+(3.75−1.5)y=1.5x+2.25y=1.5x+2.25
Now the equation is in the standard slope-intercept form, y = mx + b. By comparing our equation to this form, we can clearly identify the slope, m.
- m = 1.5
- b = 2.25
The slope (m) is 1.5. Since y is in dollars and x is in miles, this slope represents a rate of $1.50 per mile. This means that for each additional mile the taxi travels, the total cost of the ride increases by $1.50. The value 3.75 in the original equation represents the cost of the first mile, and every subsequent mile adds $1.50 to the total fare.
