Question
On a graph the cost of renting a car is based on the distance traveled
Distance in miles is x value and y is the rental cost in dollars
On graph (0,20) and (20,25)
f(x)=
Distance in miles is x value and y is the rental cost in dollars
On graph (0,20) and (20,25)
f(x)=
Answers
GPT-4o mini
To determine the function \( f(x) \) that represents the rental cost based on the distance traveled, we can observe the two points given: \( (0, 20) \) and \( (20, 25) \). Here's how we can find the linear function:
1. **Identify the points**:
- Point 1: \( (0, 20) \) means that when no distance is traveled (0 miles), the cost is $20.
- Point 2: \( (20, 25) \) means that when 20 miles are traveled, the cost is $25.
2. **Calculate the slope (m)** of the line using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here, \( (x_1, y_1) = (0, 20) \) and \( (x_2, y_2) = (20, 25) \):
\[
m = \frac{25 - 20}{20 - 0} = \frac{5}{20} = \frac{1}{4}
\]
3. **Use the point-slope form of the line**:
The point-slope form of a line is given by
\[
y - y_1 = m(x - x_1)
\]
Plugging in \( m = \frac{1}{4} \) and using the point \( (0, 20) \):
\[
y - 20 = \frac{1}{4}(x - 0)
\]
Simplifying this gives:
\[
y - 20 = \frac{1}{4}x
\]
\[
y = \frac{1}{4}x + 20
\]
4. **Final function**:
Thus the function \( f(x) \) that relates the distance traveled \( x \) to the rental cost \( f(x) \) is:
\[
f(x) = \frac{1}{4}x + 20
\]
This function indicates that the initial cost of renting the car is $20, and there is an additional charge of $0.25 for each mile traveled.
1. **Identify the points**:
- Point 1: \( (0, 20) \) means that when no distance is traveled (0 miles), the cost is $20.
- Point 2: \( (20, 25) \) means that when 20 miles are traveled, the cost is $25.
2. **Calculate the slope (m)** of the line using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here, \( (x_1, y_1) = (0, 20) \) and \( (x_2, y_2) = (20, 25) \):
\[
m = \frac{25 - 20}{20 - 0} = \frac{5}{20} = \frac{1}{4}
\]
3. **Use the point-slope form of the line**:
The point-slope form of a line is given by
\[
y - y_1 = m(x - x_1)
\]
Plugging in \( m = \frac{1}{4} \) and using the point \( (0, 20) \):
\[
y - 20 = \frac{1}{4}(x - 0)
\]
Simplifying this gives:
\[
y - 20 = \frac{1}{4}x
\]
\[
y = \frac{1}{4}x + 20
\]
4. **Final function**:
Thus the function \( f(x) \) that relates the distance traveled \( x \) to the rental cost \( f(x) \) is:
\[
f(x) = \frac{1}{4}x + 20
\]
This function indicates that the initial cost of renting the car is $20, and there is an additional charge of $0.25 for each mile traveled.
1/4x is not right
GPT-4o mini
I apologize for any confusion. Let's revisit the calculation for the linear function that represents the cost of renting a car based on the distance traveled.
Given the points:
- Point A: \( (0, 20) \)
- Point B: \( (20, 25) \)
1. **Calculate the slope (m)**: Using the formula for slope:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Plugging in the coordinates:
\[
m = \frac{25 - 20}{20 - 0} = \frac{5}{20} = \frac{1}{4}
\]
2. **Finding the equation of the line**: We know the slope \( m = \frac{1}{4} \) and we can take the first point \( (0, 20) \) to find the equation using the point-slope formula:
\[
y - y_1 = m(x - x_1)
\]
Substituting the slope and point values:
\[
y - 20 = \frac{1}{4}(x - 0)
\]
Simplifying this gives:
\[
y - 20 = \frac{1}{4}x
\]
\[
y = \frac{1}{4}x + 20
\]
3. **Final function**:
Thus, the function \( f(x) \), which gives the rental cost based on the distance \( x \), is indeed:
\[
f(x) = \frac{1}{4}x + 20
\]
If this interpretation does not match your expectations or if there’s a specific licensing agreement or aspect you want me to consider differently, please let me know, and I’ll adjust the function accordingly!
Given the points:
- Point A: \( (0, 20) \)
- Point B: \( (20, 25) \)
1. **Calculate the slope (m)**: Using the formula for slope:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Plugging in the coordinates:
\[
m = \frac{25 - 20}{20 - 0} = \frac{5}{20} = \frac{1}{4}
\]
2. **Finding the equation of the line**: We know the slope \( m = \frac{1}{4} \) and we can take the first point \( (0, 20) \) to find the equation using the point-slope formula:
\[
y - y_1 = m(x - x_1)
\]
Substituting the slope and point values:
\[
y - 20 = \frac{1}{4}(x - 0)
\]
Simplifying this gives:
\[
y - 20 = \frac{1}{4}x
\]
\[
y = \frac{1}{4}x + 20
\]
3. **Final function**:
Thus, the function \( f(x) \), which gives the rental cost based on the distance \( x \), is indeed:
\[
f(x) = \frac{1}{4}x + 20
\]
If this interpretation does not match your expectations or if there’s a specific licensing agreement or aspect you want me to consider differently, please let me know, and I’ll adjust the function accordingly!