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9 September, 01:50

A small tie shop finds that at a sales level of x ties per day its marginal profit is MP (x) dollars per tie, where MP (x) equals 1.85 plus 0.12 x minus 0.0024 x squared. Also, the shop will lose $65 per day at a sales level of x equals 0. Find the profit from operating the shop at a sales level of x ties per day.

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  1. 9 September, 02:10
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    Correct question:

    A small tie shop finds that at a sales level of x ties per day its marginal profit is MP (x) dollars per tie, where

    MP (x) = 1.85 + 0.12x - 0.0024x². Also, the shop will lose $65 per day at a sales level of x = 0. Find the profit from operating the shop at a sales level of x ties per day.

    Answer:

    P (x) = 1.85x + 0.06x² - 0.0008x³ - 65

    Step-by-step explanation:

    Given the marginal profit function:

    MP (x) = 1.85 + 0.12x - 0.0024x², P (0) = - 65

    We are to find P (x).

    P (x) = ∫MP (x) dx

    P (x) = ∫ (1.85 + 0.12x - 0.0024x²) dx

    = ∫1.85 dx+∫0.12x dx+∫ (-0.0024x²) dx + C

    = 1.85x + 0.06x² - 0.0008x³ + C

    Initial condition at P (0) = - 65

    where x (0), P (x) = - 65

    we have:

    -65 = 1.85 (0) + 0.06 (0) ² - 0.0008x (0) ³ + C

    -65 = 0 + 0 - 0 + C

    -65 = C

    C = - 65

    P (x) = 1.85x + 0.06x² - 0.0008x³ - 65
  2. 9 September, 02:14
    0
    Question:

    A small tie shop finds that at a sales level of x ties per day its marginal profit is MP (x) dollars per tie, where

    MP (x) = 1.85 + 0.12x - 0.0024x². Also, the shop will lose $65 per day at a sales level of x = 0. Find the profit from operating the shop at a sales level of x ties per day.

    Answer:

    P (x) = 1.85x + 0.06x^2 - 0.0008x^3 - 65

    Step-by-step explanation:

    MP (x) = 1.85 + 0.12 (x) - 0.0024 (x^2)

    At sales level, x=0, loss = $65

    P (0) = - $65

    Integrating MP (x) with respect to x

    P (x) = ∫MP (x) dx = ∫ 1.85 + 0.12 (x) - 0.0024 (x^2)

    P (x) = 1.85x + 0.06x^2 - 0.0008x^3 + C

    Therefore:

    P (x) = 1.85x + 0.06x^2 - 0.0008x^3 - 65
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