The monthly cost of driving a car depends on the number of miles driven. Lynn found that in May it cost her $410 to drive 300 mi and in June it cost her $470 to drive 600 mi. (a) Express the monthly cost C as a function of the distance driven, d, assuming that a linear relationship gives a suitable model. (b) Use part (a) to predict the cost of driving 1200 miles per month. (c) Draw the graph of the linear function. What does the slope represent?

Where are you stuck Ethan? Show me what you have so far and I'll point you in the right direction.

Hint: Draw a graph for yourself, designate the x axis as miles and the y as dollars. So when x is 300 what is y? And when x is 600 what is y? This might help you understand the question.

I'm stuck at part a, I don't know what the function would be

(a) To determine the linear relationship between the monthly cost C and the distance driven d, we need to find the equation of a line that passes through the two given points (300, $410) and (600, $470).

First, we find the slope of the line using the formula:

slope (m) = (change in y) / (change in x) = (470 - 410) / (600 - 300) = 60 / 300 = 1/5

The slope represents the cost of driving per mile. In this case, it is the additional cost for each mile driven.

Using the slope-intercept form of a linear equation, where y = mx + b, we can substitute one of the points to find the y-intercept (b).

Using the point (300, $410) and the slope (1/5), we have:

410 = (1/5)*300 + b
410 = 60 + b
b = 410 - 60
b = 350

Therefore, the equation for the monthly cost C as a function of the distance driven d is:

C = (1/5)d + 350

(b) To predict the cost of driving 1200 miles per month, we plug in the value of d = 1200 into the equation from part (a):

C = (1/5)*1200 + 350
C = 240 + 350
C = $590

So, the predicted cost of driving 1200 miles per month would be $590.

(c) To draw the graph of the linear function, we plot the points (300, $410) and (600, $470), and then connect them with a straight line.

The slope of the line represents the change in cost for each additional mile driven. In this case, it is $1/5 per mile. So, for every additional mile driven, the monthly cost increases by $1/5.

The y-intercept represents the fixed cost or the cost when no miles are driven. In this case, it is $350.

Therefore, the graph of the linear function will have a positive slope and will intersect the y-axis at the point (0, $350).

Note: Since only two data points are given, this linear model assumes that the relationship between the monthly cost and distance driven is linear throughout.