The monthly cost of driving a car depends on the number of miles driven. Lynn found that in May it cost her $436 to drive 320 mi, and in June it cost her $562 to drive 740 mi.
(a)
Express the monthly cost C as a function of the distance driven d, assuming that a linear relationship provides a suitable model.


To express the monthly cost $ C $ as a function of the distance driven $ d $, we can assume a linear relationship of the form:

\[
C(d) = md + b
\]

where:
- $ m $ is the slope, representing the cost per mile,
- $ b $ is the y-intercept, representing the fixed monthly cost (e.g., insurance, parking fees) that does not depend on the miles driven.

### Step 1: Set up the system of equations

Given the two points:
- $ (d_1, C_1) = (320, 436) $
- $ (d_2, C_2) = (740, 562) $

We can set up the following equations:

\[
436 = 320m + b \quad \text{(1)}
\]
\[
562 = 740m + b \quad \text{(2)}
\]

### Step 2: Solve for $ m $ (slope)

Subtract equation (1) from equation (2):

\[
562 - 436 = (740m + b) - (320m + b)
\]

\[
126 = 420m
\]

\[
m = \frac{126}{420} = 0.3
\]

So, the cost per mile $ m $ is $0.30 per mile.

### Step 3: Solve for $ b $ (intercept)

Substitute $ m = 0.3 $ into equation (1) or (2). Let's use equation (1):

\[
436 = 320(0.3) + b
\]

\[
436 = 96 + b
\]

\[
b = 436 - 96 = 340
\]

So, the fixed monthly cost $ b $ is $340.

### Final Function

The linear function that models the monthly cost $ C $ as a function of the distance driven $ d $ is:

\[
C(d) = 0.3d + 340
\]

This function suggests that for every additional mile driven, the cost increases by $0.30, and there is a fixed cost of $340 regardless of the miles driven.

Would you like more details or have any questions?

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Here are some related questions:
1. How does the fixed cost $ b $ affect the overall cost of driving?
2. If Lynn drives 500 miles in July, what would her expected monthly cost be?
3. What would the cost function look like if the cost per mile changed?
4. How would the function change if there were no fixed costs (i.e., $ b = 0 $)?
5. What does the slope $ m $ represent in a real-world context?

**Tip:** When dealing with linear functions, remember that the slope always represents the rate of change, while the intercept represents the starting value.

The monthly cost of driving a car depends on the number of miles driven. Lynn found that in May it cost her $436 to drive 320 mi, and in June it cost her $562 to drive 740 mi. (a) Express the monthly cost C as a function of the distance driven d, assuming that a linear relationship provides a suitable model.

Discover how to express the monthly car cost as a linear function of miles driven. Using real data points, learn how to calculate the cost per mile and the fixed monthly cost. This problem is suitable for students in Grades 9-11, covering key concepts in linear equations and slope-intercept form.

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