Slope Intercept Form Word Problem

Question

The speed at which you drive a car can affect the car's fuel economy. The July 2008 Consumer Reports magazine reported the Toyota Camry has a fuel economy of 40 miles per gallon (mpg) at 55 miles per hour (mph), and 30 mpg at 75 mph.

1. What is the independent and dependent variables?
2. What would be the linear equation in slope intercept form of this problem?
3. What would be a reasonable domain and range?

Hey! Sorry if it just seemed this was me getting answers or anything. I think that the independent variable would be mph and dependent would be mpg, however when I got converted into slope intercept form, my y-intercept was 67.5. I am not sure if I am going about this problem correctly and continue to solve it different ways and now I am confused whether the y intercept is 135. I am also unclear about the practical domain and range of this would be and I have to graph this problem.

Answer 1

A dependent variable is one that depends on the independent variable, usually $y$ and $x$ respectively because $x$ is the input and $y$ is the output.

That said, look at the problem's first sentence:

The speed at which you drive a car can affect the car's fuel economy.

Here, the car's fuel economy (in mpg) is affected by the speed at which you drive (mph). Thus, mpg is the dependent variable and mph is the independent variable.

Now look at the next sentence of the world problem:

The July 2008 Consumer Reports magazine reported the Toyota Camry has a fuel economy of 40 miles per gallon (mpg) at 55 miles per hour (mph), and 30 mpg at 75 mph.

Here, you're given two mph and mpg pairs for the Toyota Camry, which you can treat as ordered pairs $(55, 40)$ and $(75, 30)$. From this, you can create a linear equation by finding the slope:

$$\dfrac{30-40}{75-55} = -\dfrac{1}{2}$$

And because you know at least one point, you can rewrite into point-slope form:

$$y - 40 = -\dfrac{1}{2}(x - 55)$$

Then rearrange into slope-intercept:

$$y = -\dfrac{1}{2}x + 67.5$$

Now about a valid domain and range, think about the following questions:

• Can a car go negative mph?
• How high can modern cars' mpgs be?
• Can a car have negative mpg?

Answer 2

So you are measuring the relationship between speed $s$ and fuel usage rate $f$, and as you correctly note, $s$ seems dependent (since it is what the experiment changed) and $f$ is dependent (since this is the outcome the experiment seemed to observe).

You now have two points, to model the relationship between $s$ and $f$: $(55,40)$ and $(75,30)$,

If you assume a linear relationship between $f$ and $s$, we must say that $$f = ms + f_0, \quad \text{where } m = \frac{\Delta f}{\Delta s}.$$

Can you compute $m$ from the two points, and then use one of them to plug into the equation to find the intercept?

As for reasonable domain and range, I would not be too happy finding the quality of the model outside of the measurements.

Can you now finish this?