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Math Help - fucntions and graphs problem solving

  1. #1
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    Exclamation fucntions and graphs problem solving

    A retail furniture shop can buy dining room chairs from a manufacturer for $40 each. The retailer knows that if each chair sells for $100, 50 of them can be sold per week. However the research department believes that for every $4 reduction in price they will sell 20 more chairs per week.
    Suppose that n chairs are sold per week, that each chair sells for $x and that the total sales revenue is $R per week.

    a) show that n=550 -5x


    b) show that R= 550x -5x^2


    c) show that the profit function, P(x) = 750x -5x^2 -22000
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  2. #2
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    Hello Tessarina
    Quote Originally Posted by Tessarina View Post
    A retail furniture shop can buy dining room chairs from a manufacturer for $40 each. The retailer knows that if each chair sells for $100, 50 of them can be sold per week. However the research department believes that for every $4 reduction in price they will sell 20 more chairs per week.
    Suppose that n chairs are sold per week, that each chair sells for $x and that the total sales revenue is $R per week.

    a) show that n=550 -5x


    b) show that R= 550x -5x^2


    c) show that the profit function, P(x) = 750x -5x^2 -22000
    (a) The relationship between the number sold ( n) and the selling price ( x) is linear, because for every $4 decrease in the selling price, the number sold increases by the same amount, 20 . So, for some constants A and B:
    n = A + Bx
    When x = 100, n = 50; and when x = 96, n = 70. Therefore:
    50= A + 100B

    70 = A + 96B

    Solve these equations to find the values of A and B. Thus, the relationship between n and x is:
    n = 550-5x
    (b) The revenue from the sales is the number of chairs sold multiplied by the selling price per chair. So:
    R = nx
    Now use the answer to (a) and you're there.

    (c) Each chair costs $40 to buy. So the total cost of n chairs is $...?

    Now use part (a) to write this in terms of x.

    Then subtract this total cost from the total revenue in part (b) to find the total profit P(x) in terms of x.

    Can you complete it now?

    Grandad
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  3. #3
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    im sorry, but i still dont fully understand part c.
    i was wondering whether you could please show the working out?

    thank you very much
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  4. #4
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    Hello Tessarina
    Quote Originally Posted by Tessarina View Post
    im sorry, but i still dont fully understand part c.
    i was wondering whether you could please show the working out?

    thank you very much
    The cost of n chairs at 40 dollars per chair is 40n
    =40(550-5x), from part (a)

    =(22000 - 200x)

    Now the profit is the total revenue minus the total cost.

    So, given that the revenue is (550x - 5x^2) the profit is
    P(x) = (550x -5x^2) - (22000-200x)
    = 550x -5x^2 - 22000+200x)
    =750x-5x^2-22000
    Grandad
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  5. #5
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    I'm really sorry to ask again,
    but i was wondering whether you could also show me the whole working out for part b?

    thank you
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  6. #6
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    Hello Tessarina
    Quote Originally Posted by Tessarina View Post
    I'm really sorry to ask again,
    but i was wondering whether you could also show me the whole working out for part b?

    thank you
    The total revenue is the number of items sold multiplied by the selling price per item. If we translate this into symbols, this is:
    R = nx ...(1)
    because
    R = total revenue

    n = number of chairs sold

    x = selling price of one chair
    But we showed in part (a) that:
    n = 550 - 5x
    So in equation (1), we can replace n by (550 - 5x) to get:
    R = (550-5x)x
    and, if we remove the brackets:
    R = 550x-5x^2
    OK now?

    Grandad
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  7. #7
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    oh thank you, i now perfectly understand it all
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