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Math Help - Cost function word problems

  1. #1
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    Cost function word problems

    I have problems with these types of questions.

    A manufacturer finds it costs him x + 5x + 7 dollars to produce x tons of an item. At production levels above 3 tons, he must hire additional workers, and his costs increase by 3(x - 3) dollars on his total production. The price he receives is $13 per ton regardless of how much he manufacturers and if he has a plant capacity of 10 tons.

    a) What is the domain of the profit function?

    b) Write the profit function for a production level below 3 tons.

    c) Write the profit function for a production level above 3 tons.

    d) What level of output maximmizes profits? Justify your answer.

    so the cost function is C(x) = x + 5x + 7 if below 3 tons, but (x + 5x + 7)3(x - 3) if above 3 tons? And I'm not even sure what the last part is saying, it just throws me off.
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  2. #2
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    so the cost function is C(x) = x + 5x + 7 if below 3 tons, but (x + 5x + 7)3(x - 3) if above 3 tons?
    Not quite. You need to add the 3(x-3), not multiply by it (but that was probably just a typo). So C(x) = x^2+5x+7+3(x-3).

    a) What is the domain of the profit function?
    The profit function will be in terms of the number of tons sold. What are the possible values of tons sold?

    b) Write the profit function for a production level below 3 tons.

    c) Write the profit function for a production level above 3 tons.
    Just use Profit=Income-Cost

    d) What level of output maximmizes profits? Justify your answer.
    The easiest way to do this is probably just checking any stationary points and end points.
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