Question
A cab company charges a $4 boarding rate in addition to its meter which is $0.75 for every mile
A) What equation represents the rate of this company?
B) Graph the equation that represents the rate of this cab company?
How could we transform this graph to be proportional?
A) What equation represents the rate of this company?
B) Graph the equation that represents the rate of this cab company?
How could we transform this graph to be proportional?
Answers
Answered by
GPT-4o mini
### A) Equation Representation
The total cost \( C \) of a cab ride can be expressed as a function of the number of miles \( m \) driven. The boarding fee is a fixed charge of $4, and the variable cost is $0.75 for each mile driven. Thus, the equation that represents the rate of this company is:
\[
C(m) = 4 + 0.75m
\]
Where:
- \( C(m) \) = total cost
- \( m \) = number of miles driven
### B) Graphing the Equation
To graph the equation \( C(m) = 4 + 0.75m \):
1. **Identify the intercepts**:
- **Y-Intercept**: When \( m = 0 \):
\[
C(0) = 4 + 0.75(0) = 4
\]
Hence, the y-intercept is at the point (0, 4).
- **X-Intercept**: When \( C = 0 \):
\[
0 = 4 + 0.75m \implies 0.75m = -4 \implies m = -\frac{4}{0.75} \approx -5.33
\]
The x-intercept is not practically meaningful in the context of a cab ride, as distance cannot be negative.
2. **Slope**: The slope of the line is \( 0.75 \), which indicates that for every mile driven, the total charge increases by $0.75.
3. **Graph**:
- Start from the y-intercept at (0, 4).
- From that point, move up $0.75 for every mile to the right (the slope).
- Plot points for a few values of \( m \) (like 1 mile, 2 miles, etc.) for accuracy.
### C) Transforming the Graph to be Proportional
The current equation \( C(m) = 4 + 0.75m \) is not proportional because of the constant \( 4 \) (the boarding rate). A proportional relationship would have no fixed charge (i.e., no y-intercept other than zero) and would obey the form \( C(m) = km \), where \( k \) is a constant value.
To transform the relationship to be proportional, you would remove the fixed boarding base charge. This could be done by either eliminating the boarding fee or adjusting the business model so that it only charges per mile:
\[
C(m) = 0.75m
\]
This would make clear that the cost is proportional to the miles traveled, with no additional charge.
The total cost \( C \) of a cab ride can be expressed as a function of the number of miles \( m \) driven. The boarding fee is a fixed charge of $4, and the variable cost is $0.75 for each mile driven. Thus, the equation that represents the rate of this company is:
\[
C(m) = 4 + 0.75m
\]
Where:
- \( C(m) \) = total cost
- \( m \) = number of miles driven
### B) Graphing the Equation
To graph the equation \( C(m) = 4 + 0.75m \):
1. **Identify the intercepts**:
- **Y-Intercept**: When \( m = 0 \):
\[
C(0) = 4 + 0.75(0) = 4
\]
Hence, the y-intercept is at the point (0, 4).
- **X-Intercept**: When \( C = 0 \):
\[
0 = 4 + 0.75m \implies 0.75m = -4 \implies m = -\frac{4}{0.75} \approx -5.33
\]
The x-intercept is not practically meaningful in the context of a cab ride, as distance cannot be negative.
2. **Slope**: The slope of the line is \( 0.75 \), which indicates that for every mile driven, the total charge increases by $0.75.
3. **Graph**:
- Start from the y-intercept at (0, 4).
- From that point, move up $0.75 for every mile to the right (the slope).
- Plot points for a few values of \( m \) (like 1 mile, 2 miles, etc.) for accuracy.
### C) Transforming the Graph to be Proportional
The current equation \( C(m) = 4 + 0.75m \) is not proportional because of the constant \( 4 \) (the boarding rate). A proportional relationship would have no fixed charge (i.e., no y-intercept other than zero) and would obey the form \( C(m) = km \), where \( k \) is a constant value.
To transform the relationship to be proportional, you would remove the fixed boarding base charge. This could be done by either eliminating the boarding fee or adjusting the business model so that it only charges per mile:
\[
C(m) = 0.75m
\]
This would make clear that the cost is proportional to the miles traveled, with no additional charge.