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Quadratic Function Word Problem

  1. Feb 26, 2009 #1
    1. The problem statement, all variables and given/known data

    A company sells running shoes to dealers at a rate f $40 per pair if fewer than 50 pairs are ordered. If a dealer orders 50 or more pairs (up to 600), the price per pair is reduced at a rate of 4 cents times the number ordered. What size order will produce the maximum amount of money for the company?



    2. Relevant equations
    Vertex Formula -- -(b/2a)


    3. The attempt at a solution

    My experience from other problems like this is that the charge should be set as x and the amount of shoes sold should be y.

    In order to figure out the number of shoes sold (y), I can create two points...but I think this is where I'm confused. Initially I wanted to use (40,49) and (39.96, 51). However, the problem states the discount is a rate of 4 cents times the number ordered. For example if I ordered 100 shoes, I'd pay $36 for each one (100*.04 is 4).

    If I get two good points, I could get a slope, set up my equation for y and multiply that by x. Then I can use the vertex formula. I'm just having trouble figuring out how to write up this equation.

    Thank you!
     
  2. jcsd
  3. Feb 26, 2009 #2

    Mark44

    Staff: Mentor

    It probably makes more sense to use variable names that suggest what they're being used for. x is fine for the number of shoes ordered, but instead of y, I would suggest using C, rather than y, and with the idea that C represents the cost per pair of shoes.

    If a dealer orders x pairs of shoes, what will the cost per pair be? You'll need a function that has one definition for one set of x values, and another definition for the other set of x values.

    The revenue (R seems like a natural choice) will be the number of pairs of shoes sold times the cost per pair.
     
  4. Feb 26, 2009 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    You don't want to "figure out the number of shoes sold". You are told how to calculate the price, x, as a function of y the number of shoes sold: The "base" price is $40 per pair. If y> 50, "the price per pair is reduced at a rate of 4 cents times the number ordered" so the $40 price is reduced by 0.04y: the price for each pair of shoes is 40- 0.04y.
    The amount of money brought in is the price of each pair of shoes multiplied by the number of shoes sold: (40- 0.04y)(y)= 40y- 0.04y2. You want to find the maximum value of that, by using that "vertex formula". (I used your choice for x and y- notice that x is a quadratic function of y.) Of course, the answer must be between y= 50 and y= 600. If the vertex is not between those values the maximum will be at one of those to values.
     
  5. Feb 26, 2009 #4
    Thanks for explaining this for me. It makes sense now and it looks like 500 would produce the maximum amount of money. Thanks again!
     
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