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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.

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    $\begingroup$ Jake, welcome to Math.SE! People here usually do not like to do your homework for you. Can you please update your question with your thoughts how to approach the problem or what exactly is unclear to you about how to proceed and we will happily guide you further. $\endgroup$
    – gt6989b
    Feb 21, 2018 at 22:17
  • $\begingroup$ I apologize for the original post. I just updated it with my thoughts and what I am confused with. Thank you for the help and welcome! $\endgroup$ Feb 21, 2018 at 22:21

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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?
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  • $\begingroup$ Thank you for the answer! I am confused for the slope part because the two points I have are also (55, 40) and (75, 30), however, I got the slope of -1/2. $\endgroup$ Feb 21, 2018 at 22:36
  • $\begingroup$ I thought to find slope I would have to use the formula y2-y1/x2-x1. When I used that, I got the slope of -1/2. After converting it into point slope form I rearranged it to slope intercept, which mine was y=-1/2x+67.5. However I am really really not sure about this. $\endgroup$ Feb 21, 2018 at 22:38
  • $\begingroup$ @JakeMiller Woops, my mistake. You're right. $\endgroup$
    – Andrew Li
    Feb 21, 2018 at 22:42
  • $\begingroup$ @JakeMiller Corrected. Your answer is correct. $\endgroup$
    – Andrew Li
    Feb 21, 2018 at 22:54
  • $\begingroup$ Thank you! If I made my practical domain 0 to 100 would that make sense? And if that were the case would my practical range be 17.5 to 67.5? I'm not sure if that even makes much sense to me. I struggle with both practical domain and range. $\endgroup$ Feb 21, 2018 at 22:56
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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?

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  • $\begingroup$ Thank you for the answer! Originally, my attempted points were (55, 40) and (75, 30). I thought that since 55 and 75 were the speeds and miles per hour of each car, they would have to be independent and I thought independent variable would be x. In 40mpg at 55mph for example, since mph is the independent variable (x), would it be written (55, 40)? I am not really sure about this though and am very confused. $\endgroup$ Feb 21, 2018 at 22:33
  • $\begingroup$ @JakeMiller reread the question, updated the points. The idea is, each point is an outcome of a stand-alone experiment. So driving $55$ mph yields fuel economy of $40$ mpg, so $(55,40)$ is grouping data from one experiment. So too is grouping data from the second experiment. $\endgroup$
    – gt6989b
    Feb 21, 2018 at 22:42
  • $\begingroup$ If the independent variable is x which is the speed, how would the ordered pairs be written? I am unsure of this because I thought they would be written (x, y) which in this problem would be (mph, mpg). Would (55, 40) and (75, 30) be correct in getting the slope intercept form of y=-1/2x+67.5? Sorry for the amount of questions I am very unsure of this and myself. $\endgroup$ Feb 21, 2018 at 22:47
  • $\begingroup$ @JakeMiller yes, correct $\endgroup$
    – gt6989b
    Feb 21, 2018 at 23:12
  • $\begingroup$ thank you so much for the help! $\endgroup$ Feb 21, 2018 at 23:28

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