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Math Help - calculus application to economics

  1. #1
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    calculus application to economics

    A manufacture has been selling 1650 television sets a week at 480 each. A market survey indicates that for each 19 rebate offered to a buyer, the number of sets sold will increase by 190 per week.
    a) Find the demand function , where is the number of the television sets sold per week.

    b) How large rebate should the company offer to a buyer, in order to maximize its revenue?
    c) If the weekly cost function is , how should it set the size of the rebate to maximize its profit?
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  2. #2
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    Quote Originally Posted by xcelxp View Post
    A manufacture has been selling 1650 television sets a week at 480 each. A market survey indicates that for each 19 rebate offered to a buyer, the number of sets sold will increase by 190 per week
    ....
    b) How large rebate should the company offer to a buyer, in order to maximize its revenue?
    c) If the weekly cost function is , how should it set the size of the rebate to maximize its profit?

    Hello, xcelxp,

    to b) and c) only:

    let h be the number how often the rebate was offered, then the revenue can be calculated by:
    Code:
    r(h) = (1650 + 190h)(480 - 19h) = 
            2                   
    - 3610h  + 59850h + 792000
    The graph of this function is a parabola which opens downward that means that the maximum is the r-value of the vertex.
    Calculate the first derivative of r:
    Code:
     r'(h) = 59850 - 7220h
    
    r'(h) = 0 ==> h =  315 
                       ⎯⎯⎯
                        38
    If the rebate is 315/38 * 19 = $157.50 that means if the TV set is sold for $322.50 then the revenue has a maximum.

    to c)

    the profit is the revenue minus the cost:
    Code:
    p(h) = (1650+190h)(480-19h)-(132000+160(1650+190h))
    Expand all brackets, calculate the first derivative, let the derivative be zero, solve for h and you'll get:
    Code:
         155 
    h = ⎯⎯⎯
          38
    Plug in this value into the function p(h) to get the maximum profit: p_max = $4560625
    Number of sold TV sets: 2425 pieces.

    EB


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