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A certain company's main source of income is selling cloth bracelets. The company's annual profit (in thousands of dollars) as a function of the price of a bracelet (in dollars) is modeled by: P(x)=-2x^(2)+16 x-24 What bracelet price should the company set to earn this maximum profit? dollars

A certain company's main source of income is selling cloth bracelets.\newlineThe company's annual profit (in thousands of dollars) as a function of the price of a bracelet (in dollars) is modeled by:\newlineP(x)=−2x2+16x−24 P(x)=-2x^{2}+16x-24 \newlineWhat bracelet price should the company set to earn this maximum profit?\newlinedollars \text{dollars}

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Q. A certain company's main source of income is selling cloth bracelets.\newlineThe company's annual profit (in thousands of dollars) as a function of the price of a bracelet (in dollars) is modeled by:\newlineP(x)=−2x2+16x−24 P(x)=-2x^{2}+16x-24 \newlineWhat bracelet price should the company set to earn this maximum profit?\newlinedollars \text{dollars}
  1. Calculate x-coordinate of vertex: The profit function is given by P(x)=−2x2+16x−24P(x) = -2x^2 + 16x - 24. To find the maximum profit, we need to find the vertex of the parabola represented by this quadratic function. The x-coordinate of the vertex can be found using the formula −b2a-\frac{b}{2a}, where aa is the coefficient of x2x^2 and bb is the coefficient of xx.
  2. Identify coefficients: In the given function P(x)=−2x2+16x−24P(x) = -2x^2 + 16x - 24, the coefficient aa is −2-2 and the coefficient bb is 1616. Let's calculate the x-coordinate of the vertex using the formula −b2a-\frac{b}{2a}.x=−b2a=−162∗(−2)=−16−4=4x = -\frac{b}{2a} = -\frac{16}{2*(-2)} = -\frac{16}{-4} = 4.
  3. Determine optimal price: The xx-coordinate of the vertex is 44, which means that the company should set the price of a bracelet at $4\$4 to earn the maximum profit.
  4. Verify maximum profit: To ensure that this price indeed gives the maximum profit, we can check the coefficient of the x2x^2 term in the profit function. Since the coefficient is −2-2, which is negative, the parabola opens downwards, confirming that the vertex represents the maximum point.

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