Answer :
To analyze what the slope means in the context of the given equation for the cost of a taxi ride, [tex]$c(x)=2x+3.00$[/tex], let's break down the equation and interpret its components correctly.
The equation [tex]$c(x) = 2x + 3.00$[/tex] represents a linear relationship between the cost \(c(x)\) and the number of minutes \(x\) for a taxi ride. In such a linear equation, the general form is \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept.
1. Identify the slope and the y-intercept:
- Here, \(2\) is the coefficient of \(x\).
- The slope \(m\) is \(2\).
- The y-intercept \(b\) is \(3.00\).
2. Interpret the slope in the context of the problem:
- The slope \(m\) represents the rate of change of the cost with respect to time \(x\) (number of minutes).
- Specifically, it tells us how much the cost will increase for each additional minute of the taxi ride.
3. Determine the meaning of the slope:
- Since the slope \(m = 2\), this indicates that the cost of the taxi ride increases by \(\$2.00\) for each additional minute.
Given this interpretation, we can break down the given multiple-choice options:
- A. The taxi ride costs a total of [tex]$\$[/tex] 3.00$.
- This option is incorrect because [tex]$3.00$[/tex] is the y-intercept, which represents the initial fee or fixed cost for the taxi ride, not the slope.
- B. The rate of change of the cost of the taxi ride is [tex]$\$[/tex] 2.00$ per minute.
- This option is correct because the slope \(2\) indicates the rate at which the cost increases per minute.
- C. The taxi ride costs [tex]$\$[/tex] 2.00$ per trip.
- This option is incorrect because this statement misinterprets the slope as a fixed cost per trip, rather than the rate per minute.
- D. The rate of change of the cost of the taxi ride is [tex]$\$[/tex] 3.00$ per minute.
- This option is incorrect because it misstates the slope. The correct rate of change is [tex]$\$[/tex]2.00[tex]$ per minute, not $[/tex]\[tex]$3.00$[/tex].
Therefore, the correct interpretation of the slope, based on the given linear equation \(c(x)=2x+3.00\), is:
B. The rate of change of the cost of the taxi ride is [tex]$\$[/tex] 2.00$ per minute.
The equation [tex]$c(x) = 2x + 3.00$[/tex] represents a linear relationship between the cost \(c(x)\) and the number of minutes \(x\) for a taxi ride. In such a linear equation, the general form is \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept.
1. Identify the slope and the y-intercept:
- Here, \(2\) is the coefficient of \(x\).
- The slope \(m\) is \(2\).
- The y-intercept \(b\) is \(3.00\).
2. Interpret the slope in the context of the problem:
- The slope \(m\) represents the rate of change of the cost with respect to time \(x\) (number of minutes).
- Specifically, it tells us how much the cost will increase for each additional minute of the taxi ride.
3. Determine the meaning of the slope:
- Since the slope \(m = 2\), this indicates that the cost of the taxi ride increases by \(\$2.00\) for each additional minute.
Given this interpretation, we can break down the given multiple-choice options:
- A. The taxi ride costs a total of [tex]$\$[/tex] 3.00$.
- This option is incorrect because [tex]$3.00$[/tex] is the y-intercept, which represents the initial fee or fixed cost for the taxi ride, not the slope.
- B. The rate of change of the cost of the taxi ride is [tex]$\$[/tex] 2.00$ per minute.
- This option is correct because the slope \(2\) indicates the rate at which the cost increases per minute.
- C. The taxi ride costs [tex]$\$[/tex] 2.00$ per trip.
- This option is incorrect because this statement misinterprets the slope as a fixed cost per trip, rather than the rate per minute.
- D. The rate of change of the cost of the taxi ride is [tex]$\$[/tex] 3.00$ per minute.
- This option is incorrect because it misstates the slope. The correct rate of change is [tex]$\$[/tex]2.00[tex]$ per minute, not $[/tex]\[tex]$3.00$[/tex].
Therefore, the correct interpretation of the slope, based on the given linear equation \(c(x)=2x+3.00\), is:
B. The rate of change of the cost of the taxi ride is [tex]$\$[/tex] 2.00$ per minute.
