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Math Help - Optimization...

  1. #1
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    Optimization...

    Ok so this one is hanging me up...

    Two industrial plants, A and B, are located 20 miles apart and emit 40 ppm (parts per million)
    and 90 ppm of particulate matter, respectively. Each plant is surrounded by a restricted area
    of radius 1 mile in which no housing is allowed and the concentration of pollutant arriving at
    any other point Q from each plant decreases with the reciprocal of the distance between that
    plant and Q. A warehouse need to be built on a road joining the two plants.

    a) Lets say the house is located x miles from plant A (and hence 20 x from plant B). Express
    the total concentration of particulate matter arriving from both plants. What is the condition
    on x?
    b) Where should the house be located on the road to minimize the total pollution arriving
    from both plants? (That is, find x to minimize the function you have in question a )
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  2. #2
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    Quote Originally Posted by jjoechump View Post
    Ok so this one is hanging me up...

    Two industrial plants, A and B, are located 20 miles apart and emit 40 ppm (parts per million)
    and 90 ppm of particulate matter, respectively. Each plant is surrounded by a restricted area
    of radius 1 mile in which no housing is allowed and the concentration of pollutant arriving at
    any other point Q from each plant decreases with the reciprocal of the distance between that
    plant and Q. A warehouse need to be built on a road joining the two plants.

    a) Lets say the house is located x miles from plant A (and hence 20 x from plant B). Express
    the total concentration of particulate matter arriving from both plants. What is the condition
    on x?
    b) Where should the house be located on the road to minimize the total pollution arriving
    from both plants? (That is, find x to minimize the function you have in question a )
    to a):

    Let m denote the amount of matter then m is calculated by:

    m(x)=\dfrac1x \cdot 40\ ppm + \dfrac1{20-x} \cdot 90\ ppm = \dfrac{50(x+16)}{20x-x^2} ppm~,~ 1 < x < 19

    to b)

    Determine the first derivation (use quotient rule) and solve the equation m'(x) = 0 for x.

    (For your confirmation only: x = -40 or x = 8. Keep in mind that there are some restrictions on x!)
    Attached Thumbnails Attached Thumbnails Optimization...-min_pollution.png  
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