The function C(x)=−20x+1681 represents the cost to produce x items. What is the least number of items that can be produced so that the avera

The function C(x)=−20x+1681 represents the cost to produce x items. What is the least number of items that can be produced so that the average cost is no more than $21?

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  1. Answer:

    The least number of items to produce is 41

    Step-by-step explanation:

    Average Cost

    Given C(x) as the cost function to produce x items. The average cost is:

    [tex]\displaystyle \bar C(X)=\frac{C(x)}{x}[/tex]

    The cost function is:

    [tex]C(x) = -20x+1681[/tex]

    And the average cost function is:

    [tex]\displaystyle \bar C(X)=\frac{-20x+1681}{x}[/tex]

    We are required to find the least number of items that can be produced so the average cost is less or equal to $21.

    We set the inequality:

    [tex]\displaystyle \frac{-20x+1681}{x}\le 21[/tex]

    Multiplying by x:

    [tex]-20x+1681 \le 21x[/tex]

    Note we multiplied by x and did not flip the inequality sign because its value cannot be negative.

    Adding 20x:

    [tex]1681 \le 21x+20x[/tex]

    [tex]1681 \le 41x[/tex]

    Swapping sides and changing the sign:

    [tex]41x \ge 1681[/tex]

    Dividing by 41:

    [tex]x\ge 41[/tex]

    The least number of items to produce is 41

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