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Writing equations and calculating cost based on a graph

  1. Apr 20, 2016 #1
    1. The problem statement, all variables and given/known data
    12. The Doggone Leash Company offers personalized dog accessories. The dog’s name is printed on items such as leashes and collars. A basic fee is charged for the first 10 letters of the dog’s name, and an additional charge applies to each letter after that. The graph shows the relationship between the total cost of the service and the number of additional letters in the dog’s name.
    upload_2016-4-20_14-46-47.png

    a. Write an equation that models this relationship and define the variables that you use.
    b.What is the cost of each additional letter? Explain how you know.
    c.What is the cost of a leash with no additional letters? Explain how you know.
    d. Create a table of values that shows the cost of a leash for 0 to 5 extra letters.

    2. Relevant equations

    y=mx+b

    3. The attempt at a solution
    so here are my answers, are they correct?

    a. Write an equation that models this relationship and define the variables that you use.
    In the equation y = mx + b, y represents the total cost of personalized dog accessories, m represents the basic fee, x represents the number of 10 letters, and b represents the additional fee. According to the graph the basic fee is $11. I also noticed every time an additional letter is added the cost increases by $1, meaning the additional fee is $1.
    An equation that models this relationship is:

    y = 11x + 1



    b. What is the cost of each additional letter? Explain how you know.
    By looking at the graph I can tell that the cost is increasing by $1 for each additional letter added on. This indicates that the cost of an additional letter is $1.


    c. What is the cost of a leash with no additional letters? Explain how you know.
    Since there are no additional letters the amount of letter purchased is 10. The basic fee for 10 letters is $11. Therefore, the cost of a leash is $11.


    d. Create a table of values that shows the cost of a leash for 0 to 5 extra letters.
    upload_2016-4-20_14-48-39.png
     
  2. jcsd
  3. Apr 20, 2016 #2

    Math_QED

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    The equation of the line is wrong. According to the graph, when x = 0, you don't have additional letters, it costs 11$. When I plug x = 0 in your equation, it would cost 1$.

    Do you know what the meaning is of m? And how m is defined?
     
  4. Apr 20, 2016 #3
    by m you mean slope? As in the formula y=mx+b?
     
  5. Apr 20, 2016 #4

    Math_QED

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  6. Apr 20, 2016 #5
    wait so the m value i got is wrong? would it perhaps be y =10x+1?
     
  7. Apr 20, 2016 #6
    Oh I get what you mean when you do 11 x 0 you get 0 then add 1 so the total would be $1. So how would i fix the equation? Wait could i do y= 1x +11? If you subisitiue x for 0 it would work
     
  8. Apr 20, 2016 #7

    SammyS

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    That gives the correct y-intercept.

    However, it looks like you are just guessing at the slope.
     
    Last edited: Apr 21, 2016
  9. Apr 21, 2016 #8

    Math_QED

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    Yes, he seems to be guessing the slope. I repeat my question for the OP. Do you know what the m in y = mx + b represent? And how it is mathematically defined?
     
  10. Apr 21, 2016 #9

    Ray Vickson

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    Your table is almost completely wrong. Look at the graph: for 3 additional letters the cost is not the $14 that you say, and for 5 additional letters the cost is not $17. Where on earth did you get these numbers?

    If you look for two (correct) points on the graph, that will be enough to allow you to find the line easily, because in other threads you already solved such problems, and you just need to do the same thing here.
     
  11. Apr 21, 2016 #10
    y = mx+b is the equations for a line, m is the slope, b is the y intercept and y is the whole line. No I didn't guess I thought that I put the values into the equations wrong. so i switched y and b so that they make sense. O I think I know why you said it was incorrect. The basic fee doesn't relate with the additional fee and extra letters. Since the additional fee depends on how many extra letters you have that means they are being multiplied so I'm pretty sure that this equation would be correct and I tried it and it works : y = 1x + 11

    So I think i found my error. I hope its correct :)
     
  12. Apr 21, 2016 #11

    SammyS

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    Your slope is wrong. That's probably related to the fact that the numbers in your table are wrong.

    Two very definite (easy to read - no fraction part) points on your graph occur for 4 extra letters and for 8 extra letters. The extra cost (above $11) is not $4 and $8 in those two cases.

    The slope is NOT 1 .
     
  13. Apr 21, 2016 #12

    ok I will recalculate the slope. I knew what I was doing the other few questions but for this one I went blank. so its completely wrong, I'm glad I posted this on PF or I would have lost some good marks.
     
  14. Apr 21, 2016 #13

    SammyS

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    Just look at rise/run .
     
  15. Apr 21, 2016 #14
    Ok I will and will post the new slope soon.
     
  16. Apr 21, 2016 #15

    Math_QED

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    You didn't yet answer the question. What is m mathematically, and do not say 'slope'. Since I don't think you know what this means, I will give you both a mathematical and intuitive description.

    Intuitive

    When you go 1 unit to the right, you go m up (m positive) or m down (m negative)

    Mathematical

    When you have 2 points P and Q with coordinates (x1,y1) and (x2,y2) that are part of the line, you can describe this line using these 2 points, because we can only draw one unique line through 2 different points (Euclid's axiom). Because m represents how fast your line 'rises', m = (y2 - y1)/(x2 - x1). When you have had physics courses about average speed, you saw a formula v = x/t. This may help you to understand the concept.
     
  17. Apr 22, 2016 #16
    upload_2016-4-22_15-44-52.png

    So i picked the 2 points (1,12) and (4,16). So I did y2 - y1/ x2 - x1. Which is 4 - 1/ 16 - 12. I got the new slope 4/3. Which is 1.3, but that wont make sense. I still don't get how im wrong?
     
  18. Apr 22, 2016 #17

    Math_QED

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    When I first looked at the graph, it seemed like the slope was 1. But a closer look says me that it's not 1. Note that when you look very closely, the coordinate (1,12) is not a part of the line. So you might want to take another point on the line, calculate m, and find the correct equation of the line.
     
  19. Apr 22, 2016 #18

    SammyS

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    Look more carefully at your graph.

    By sating that the graph goes through (1, 12), you are saying that 1 additional letter costs one additional dollar over the $11. If you use this rounded off value of slope (1.3), one additional letter costs $1.30 over the $11.

    Careful observation of the graph shows that the graph passes through the $12 mark at a bit less that 1 extra letter and passes the one additional letter mark at a bit over $12.

    You have already been given STRONG hints to look at the 4 extra letter mark and the 8 extra letter mark.

    What other point should you know with confidence?
     
  20. Apr 22, 2016 #19
    O i see i zoomed in the graph all the way. And it seems that the point 1,12 was 1, 12.2 (or something close to that). Ok now i picked 2 points that aren't decimals. The 2 points are (4,16) and (8,21). I did the y2-y1 / x2-x1 and got the slope of 5/2, which is 2.5. Is that slope correct?
     
  21. Apr 22, 2016 #20

    SammyS

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    Check your arithmetic .
     
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