The total cost (in dollars) for a company to manufacture and sell x items per week is C=40x+180, whereas the revenue brought in by selling a

Question

The total cost (in dollars) for a company to manufacture and sell x items per week is C=40x+180, whereas the revenue brought in by selling all x items is R=68x−0.4×2. How many items must be sold to obtain a weekly profit of $300?

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Mộc Miên 5 months 2021-08-31T05:46:58+00:00 1 Answers 7 views 0

Answers ( )

  1. diemkieu
    0
    2021-08-31T05:48:55+00:00
    Reply

    Answer:

    The company needs to sell either 30 or 40 items.

    Step-by-step explanation:

    We are given that the cost for selling x items given by the function:

    C(x)=40x+180

    And the revenue for selling x items is given by:

    R(x)=68x-0.4x^2

    The profit function is the cost function subtracted from the revenue function:

    P(x)=R(x)-C(x)

    Substitute and simplify:

    \displaystyle \begin{aligned} P(x)&=(68x-0.4x^2)-(40x+180)\\&=68x-0.4x^2-40x-180\\&=-0.4x^2+28x-180\end{aligned}

    To find how many items must be sold in order to obtain a weekly profit of $300, we can let P equal 300 and solve for x. So:

    300=-0.4x^2+28x-180

    Solve for x. Subtract 300 from both sides:

    -0.4x^2+28x-480=0

    We can divide both sides by -0.4:

    x^2-70x+1200=0

    Factor:

    (x-40)(x-30)=0

    Zero Product Property:

    x-40=0\text{ or } x-30=0

    Solve for each case:

    x=40\text{ or } x=30

    So, in order to obtain a weekly profit of $300, the company need to sell either 30 or 40 items.

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