Consider the cost function for a taxi ride given by [tex]\( c(x) = 2x + 3.00 \)[/tex], where [tex]\( c(x) \)[/tex] represents the cost in dollars and [tex]\( x \)[/tex] represents the number of minutes.
The general form of a linear equation is [tex]\( y = mx + b \)[/tex], where:
- [tex]\( m \)[/tex] is the slope of the line, representing the rate of change of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex].
- [tex]\( b \)[/tex] is the y-intercept, representing the value of [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is zero.
In the given cost function [tex]\( c(x) = 2x + 3.00 \)[/tex]:
- [tex]\( m = 2 \)[/tex], which is the slope,
- [tex]\( b = 3.00 \)[/tex], which is the y-intercept.
Let's interpret what these values mean in the context of the taxi ride.
1. Slope ([tex]\( m = 2 \)[/tex]):
- The slope represents the rate of change of the cost with respect to the time. Specifically, it indicates how much the cost increases for each additional minute.
- Here, the slope is 2, which means for every additional minute [tex]\( x \)[/tex], the cost [tex]\( c(x) \)[/tex] increases by [tex]$2.00.
- Therefore, the correct interpretation of the slope is that the cost of the taxi ride increases by $[/tex]2.00 per minute.
2. Y-intercept ([tex]\( b = 3.00 \)[/tex]):
- The y-intercept represents the initial cost of the ride before any time has accrued.
- At [tex]\( x = 0 \)[/tex] (i.e., at the start of the trip), the cost [tex]\( c(0) = 3.00 \)[/tex].
- This means that there is a base fare or initial charge of [tex]$3.00 just for getting into the taxi.
Given this explanation, the correct interpretation of the slope in this situation is:
A. The rate of change of the cost of the taxi ride is $[/tex]2.00 per minute.
Therefore, the answer is:
A. The rate of change of the cost of the taxi ride is $2.00 per minute.
