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Math Help - Dishwashing mathmatical help

  1. #1
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    Dishwashing mathmatical help needed!

    Hey, I'm new here, because I dont no where else to go with my bemusing problem...

    I'm taking part in a Science Technology Engineering and Mathmatics course, but this problem which I came across is something I haven't covered within my skillset yet...

    Firstly, the basic problem is about Re-usable cups vs Disposable cups, but I'm having problems doing a certian energy calculation.

    I worked out that the energy cost for a re-usable cup would be:
    n/x + x(r) with x being number of uses/washes, r being the energy to wash a cup in a dishwasher in joules (total dishwasher energy/total number of cups) and n being the production energy, also in joules. However, I know some of these values, making the problem:

    6,300,000/x + x(64,800) = the total energy (which I will call T)


    Now what I got stuck on is this...


    I know it takes 310,000 joules of energy to make a plastic disposable cup. I want to find out how many times I need to use and wash the cup to get a payback time on my re-using.


    So I got T=6,300,000/x + x(64,800)
    But with T being 310,000


    310,000 = 6,300,000/x + 64800x


    But how do I find x?


    I end up with:


    6,300,000/x + x(64,800) = 310,000

    6,300,000/x + 64,800x = 310,000

    64,800x + 6,300,000/x = 310,000

    64,800x = 310,000 - 6,300,000/x


    and I just stop...

    Have I gone wrong somewhere?
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  2. #2
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    i have not checked your derivation, but you can solve the equation you got stuck on by multiplying both sides by x.
    This gives
    64,800x^2 = 310,000x - 6,300,000
    64,800x^2 - 310,000x + 6,300,000 = 0

    This is a quadratic equation that you can solve using the quadratic formula.
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  3. #3
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    So I put that into a quadratic calculator (because I dont know how to do them) and...

    Input MUST have the format:
    AX2 + BX + C = 0

    EXAMPLE: input the equation
    2X2 + 4X -30 = 0 as:
    A= 2 B= 4 C= -30

    click E N T E R and your answers should be 3 and -5.


    Note: Do not enter commas.

    A= 64800 B= -310000 C= 6300000


    X1= 2.392 plus 9.5656 i
    X2= 2.392 minus 9.5656 i
    Here's the answer in fractional form:
    X= { 310000 Sq Root ( 9.6100e+10 - 1.6330e+12 ) } / 129600

    From here, I'm not sure how to get to an answer of how many times I need to re-use a cup and wash it to equal 310,000 joules of energy per use.
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  4. #4
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    Those outputs mean that the quadratic equation you derived has no solution in real numbers. In the context of this problem, it means that the equation you derived is not correct.


    Lets start over from the beginning.

    Energy to produce reusable cup: 6,300,000
    Energy to wash reusable cup: 64,800

    I'll assume you wash it before using it the first time.

    Find the Total energy used once you have used the cup x times. This is the production cost plus the cost of washing it x times.
    6,300,000 + 64,800x



    Now find the total energy needed to use x disposable cups
    310,000x


    All you want to do is find the value of x that makes these two figures equal.
    310,000x = 6,300,000 + 64,800x
    Can you solve this equation for x?

    Spoiler:

    Take 64800x from both sides

    310,000x - 64800x = 6,300,000
    246,000x = 6,300,000
    x =  25.609

    so you will be better off if you use the cup at least 26 times

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  5. #5
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    Quote Originally Posted by SpringFan25 View Post
    Those outputs mean that the quadratic equation you derived has no solution in real numbers. In the context of this problem, it means that the equation you derived is not correct.


    Lets start over from the beginning.

    Energy to produce reusable cup: 6,300,000
    Energy to wash reusable cup: 64,800

    I'll assume you wash it before using it the first time.

    Find the Total energy used once you have used the cup x times. This is the production cost plus the cost of washing it x times.
    6,300,000 + 64,800x



    Now find the total energy needed to use x disposable cups
    310,000x


    All you want to do is find the value of x that makes these two figures equal.
    310,000x = 6,300,000 + 64,800x
    Can you solve this equation for x?
    I'm not quite sure what you mean sorry...

    I'll show you what I've done so far:

    After 1 use:

    n = 6,300,000 and x= 1
    (6,300,000/1) + 1(64800)
    Total energy = 6,364,800

    After 2 uses:



    n = 6,300,000 and x = 2
    (6,300,000/2) + 2(64800)
    Total energy = 3,279,600


    After 5 uses:

    n = 6,300,000 and x= 5
    (6,300,000/5) + 5(64800)
    Total energy = 1,584,000


    I want to solve the problem: If total energy = the energy it takes to produce a disposable cup (310,000), what is x (how many times must I use a re-usable cup for it to match the energy cost for 1 use of a disposable cup)

    I'm pretty sure I should divide something which I haven't...
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  6. #6
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    it is not correct to divide 6,300,000 by x and multiply 68400 by x. Depending on how you approach the question, you should do one or the other.

    6,300,000 is the production cost of the cup, and it is constant. it doesn't decrease because you washed the cup 1 more time.




    I dont think you have written the question correctly either
    how many times must I use a re-usable cup for it to match the energy cost for 1 use of a disposable cup)
    Do you mean
    how many times must I use a re-usable cup for the average energy cost to match the energy cost for 1 use of a disposable cup?


    in this case, the average cost per use of the disposable cup is
    \frac{6,300,000}{x} + \frac{64800x}{x}

    = \frac{6,300,000}{x} +64800

    not \frac{6,300,000}{x} + 64800x
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  7. #7
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    Quote Originally Posted by SpringFan25 View Post
    it is not correct to divide 6,300,000 by x and multiply 68400 by x. Depending on how you approach the question, you should do one or the other.

    6,300,000 is the production cost of the cup, and it is constant. it doesn't decrease because you washed the cup 1 more time.




    I dont think you have written the question correctly either


    Do you mean
    how many times must I use a re-usable cup for the average energy cost to match the energy cost for 1 use of a disposable cup?


    in this case, the average cost per use of the disposable cup is
    \frac{6,300,000}{x} + \frac{64800x}{x}

    = \frac{6,300,000}{x} +64800

    not \frac{6,300,000}{x} + 64800x
    Yes, thanks, I've realised my mistake now.

    [n + x(64,800)]/x with x being number of uses/washes, and n being the production energy

    Cancel to 6300000/x+ 64800
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  8. #8
    Senior Member Mukilab's Avatar
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    Shouldn't it be

    'Struggling Mathematician'?
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