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Math Help - Trying to solve Alg 2 problem

  1. #1
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    Trying to solve Alg 2 problem

    I'm trying to help a student with a word problem and am stuck on it. He's learning how to use functions to find maximum and minimum values for word problems. None of the examples in his text are like the problem below, and we can't figure it out.:


    "A publishing company can get 500 subscribers for a weekly news magazine if the yearly subscription rate is $60. The company will get 10 more subscribers for each $1 decrease in the yearly rate. What rate will produce the maximum income, and what will that income be?"


    In other problems of this type, we'd say the following:
    x = additional subscribers
    500 + x = total subscribers
    60-x = subscription rate
    Then if you solve for the f(x) = (500+x)(60-x) you get an equation that tells the maximum income and also the maximum income. But, in this case it's not $1 per additional subscriber, but per 10 subscribers. What am I doing wrong in my equation then?
    Thanks!!
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  2. #2
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    Quote Originally Posted by hereami View Post
    I'm trying to help a student with a word problem and am stuck on it. He's learning how to use functions to find maximum and minimum values for word problems. None of the examples in his text are like the problem below, and we can't figure it out.:


    "A publishing company can get 500 subscribers for a weekly news magazine if the yearly subscription rate is $60. The company will get 10 more subscribers for each $1 decrease in the yearly rate. What rate will produce the maximum income, and what will that income be?"


    In other problems of this type, we'd say the following:
    x = additional subscribers
    500 + x = total subscribers
    60-x = subscription rate
    Then if you solve for the f(x) = (500+x)(60-x) you get an equation that tells the maximum income and also the maximum income. But, in this case it's not $1 per additional subscriber, but per 10 subscribers. What am I doing wrong in my equation then?
    Thanks!!
    let x = number of $1 decreases in the subscription rate

    500+10x = number of subscribers

    60-x = subscription rate

    income ... f(x) = (500+10x)(60-x)

    max income at x = 5

    max income ... f(5) = (550)(55) = $30250
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  3. #3
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    still unsure...

    Okay, so I understand how you set it up. But, how did you solve f(x) = (500+10x)(60-x)?

    I get f(x) = 30,000 - 500x +600x -10x 2
    =-10x 2 +100x + 30,000
    = x 2-10x - 3,000

    Right so far? Then if I complete the square I get
    =(x-5)2 -3025.
    But that's not what you ended up with. What am I missing?
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  4. #4
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    the function is quadratic, and its graph is a parabola that opens down.

    setting f(x) = 0 ...

    (500+10x)(60-x) = 0

    x-intercepts at x = -50 and x = 60

    because of the parabola's symmetry, the vertex is located midway between the two x-intercepts ... at x = 5.

    that's where f(x) has its maximum.
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