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Math Help - Optimization Problem.

  1. #1
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    Optimization Problem.

    A farmer wants to hire workers to pick 900 bushels of beans. Each worker can pick 5 bushels per hour and is paid $2 per bushel. The farmer must also hire a supervisor at $10 per hour while the picking is in progress, and has additional costs of $8 for each worker hired. How many workers should he hire in order to minimize the total cost? What will be his total costs? Draw a graph of the cost function.

    Not sure how to procede on this one. There seems to be some variables missing like, how long does he have to pick the beans (is it per day?). Based on the assumption, there only needs to be one but has to be at least one supervisor. So that ($10 + $8x)t for the cost of the workers and ($2*5)t profit. Not sure where to go from here.
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  2. #2
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    Quote Originally Posted by ffuh2 View Post
    A farmer wants to hire workers to pick 900 bushels of beans. Each worker can pick 5 bushels per hour and is paid $2 per bushel. The farmer must also hire a supervisor at $10 per hour while the picking is in progress, and has additional costs of $8 for each worker hired. How many workers should he hire in order to minimize the total cost? What will be his total costs? Draw a graph of the cost function.

    Not sure how to procede on this one. There seems to be some variables missing like, how long does he have to pick the beans (is it per day?). Based on the assumption, there only needs to be one but has to be at least one supervisor. So that ($10 + $8x)t for the cost of the workers and ($2*5)t profit. Not sure where to go from here.
    Presumably they pick until they are finished picking the 900 bushels! I can't see why there should be any limit on time except that. If there are x workers and each can pick 5 bushels per hour, then, since 900/5= 180, this will require 180/x hours. Multiply that by $10 (since each worker is paid $2 per bushel and picks 5 bushels per hour) to find the wages- add 8x for the "additional cost" and 10(180/x) for the supervisor.
    (I am assuming that the "supervisor" is not included in "workers".)

    Since it says to graph that anyway, go ahead and graph on, say, a graphing calculator to find the minimum value.

    Or take the derivative with respect to x and set that equal to 0.
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