The monthly cost of driving a car depends on the number of...

The monthly cost of driving a car depends on the number of miles driven. Lynn found that in May it cost her $\$ 380$ to drive 480 mi and in June it cost her $\$ 460$ to drive 800 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 1500 miles per month. (c) Draw the graph of the linear function. What does the slope represent? (d) What does the $y$ -intercept represent? (e) Why does a linear function give a suitable model in this situation?

The monthly cost of driving a car depends on the number of
miles driven. Lynn found that in May it cost her $\$ 380$ to
drive 480 mi and in June it cost her $\$ 460$ to drive 800 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 1500 miles per
month.
(c) Draw the graph of the linear function. What does the
slope represent?
(d) What does the $y$ -intercept represent?
(e) Why does a linear function give a suitable model in this
situation?

Key Concepts

Graphing Linear Functions

Graphing a linear function provides a visual representation of the relationship between variables. The straight-line graph clearly shows the constant rate of change (slope) and the starting value (y-intercept), which helps in understanding and interpreting the behavior of the modeled situation.

Prediction Using a Linear Function

Prediction using a linear function involves substituting a specific value into the linear equation to forecast an outcome. This technique is essential in many applications, enabling the estimation of unknown values based on observed trends in data.

Linear Modeling

Linear modeling involves using a straight-line equation to represent and analyze relationships between variables. This method is useful in real-world situations where costs or other quantities change at a constant rate with respect to another variable, allowing for simple predictions and interpretations.

Y-intercept

The y-intercept is the value of the dependent variable when the independent variable is zero. In a cost function, the y-intercept indicates the fixed cost that is incurred even when no distance is driven, highlighting baseline expenses that do not depend on usage.

Slope

The slope in a linear function represents the rate of change between the two variables. In cost modeling, it quantifies how much the cost increases for each additional unit of distance driven. This is a key concept because it shows the marginal cost per mile.

Linear Function

A linear function is a mathematical relationship where the change in one variable is proportional to the change in another. In the context of cost modeling, it means that as the distance driven increases, the cost increases at a constant rate, resulting in a straight-line graph when plotted.

Transcript

00:02 In this problem, we are told that the monthly cost of driving a car is a function of the amount of miles driven, and we are given some data points which relate the monthly cost of driving a car and the number of miles driven.

00:30 In part a, we're asked to find a function which relates these two.

00:37 So let's call c our monthly cost, and then let's say d is the number of miles driven.

00:54 So c of d is the monthly cost given d miles have been driven.

01:05 And we're going to assume that a linear relationship exists so that c of d is going to be equal to, mx plus b for some m and b in the real numbers now we're given that after driving 480 miles it costs three hundred eighty dollars for the month of may so in one month after driving forty miles we see that the cost is 380 and in another month june after driving 800 miles, the monthly cost was 460.

02:14 Therefore, we have two points that determine a line.

02:22 So to find the slope, we can simply think of the slope as change in y over change in x.

02:31 So we can write m equals 460 minus 380 over 800 minus 480, which is equal to 1.

03:06 1 over.

03:18 So we have in particular c of 480 equal to 1 quarter times 480 plus b, which is then equal to 380.

03:39 So we have that b is equal to 380 minus a quarter of 480, which is 380 minus 120, which is in turn equal to 260.

04:00 So we get that c of d is equal to 1 quarter d plus 260.

04:20 And this is the answer for part a.

04:25 In part b, we are asked to use the function to make a prediction.

04:36 So suppose the cost of driving is given by this form formula, and we want to know if we were to drive 1 ,500 miles, what would the cost be? well, simply evaluate c of 1500 is equal to a quarter times 1500 plus 260, which is equal to, let's see, we have 3, 375, a bit of mental math there, plus 260.

05:27 Plus 260, which is equal to 500 plus 130, which is 630, which is 630 plus 5, 635.

05:43 And this is in dollars per month.

05:54 And this is our answer for part b.

06:12 Part c, we're asked to graph the function and interpret the slope.

06:21 So, and we also know that cost is going to be strictly non -negative because it doesn't seem to make sense that they would a rental company would pay you to drive a car some distance that's not how it works so you would have to pay that much so therefore the cost is always greater than equal to zero so this is all taking place in the first quadrant and we know that our formula is c of d is a quarter d plus 260 so we have y intercept at 260 and we were we're also given a couple points.

07:27 In particular, c of 480 is 380.

07:33 So we say this is 480 here.

07:44 We can draw another point and connect them with a line.

07:47 And this is our function, this graph.

07:52 Now the slope is equal to 1 over 4.

07:57 And we'll interpret that as the change in the cost of renting the car with respect to the change in the amount of miles driven.

08:19 In other words, it's actually still an increase, even though it's a very small increase...