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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 is the maximum profit that the company can earn? thousand 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−24P(x)=-2x^{2}+16x-24\newlineWhat is the maximum profit that the company can earn?\newlinethousand 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−24P(x)=-2x^{2}+16x-24\newlineWhat is the maximum profit that the company can earn?\newlinethousand dollars
  1. Identify profit function: Identify the profit function.\newlineThe profit function is given by P(x)=−2x2+16x−24P(x) = -2x^2 + 16x - 24, where P(x)P(x) is the profit in thousands of dollars and xx is the price of a bracelet in dollars.
  2. Recognize quadratic equation form: Recognize that the profit function is a quadratic equation in the form of P(x)=ax2+bx+cP(x) = ax^2 + bx + c, where a=−2a = -2, b=16b = 16, and c=−24c = -24.\newlineSince the coefficient of x2x^2 is negative (a=−2a = -2), the parabola opens downward, which means the vertex of the parabola will give the maximum profit.
  3. Calculate vertex x-coordinate: Calculate the x-coordinate of the vertex using the formula x=−b2ax = -\frac{b}{2a}.\newlineFor the given function, a=−2a = -2 and b=16b = 16, so x=−162⋅−2=−16−4=4x = -\frac{16}{2 \cdot -2} = -\frac{16}{-4} = 4.
  4. Substitute x-coordinate into profit function: Substitute the x-coordinate back into the profit function to find the maximum profit.\newlineP(4)=−2(4)2+16(4)−24=−2(16)+64−24=−32+64−24=32−24=8P(4) = -2(4)^2 + 16(4) - 24 = -2(16) + 64 - 24 = -32 + 64 - 24 = 32 - 24 = 8.
  5. Interpret the result: Interpret the result.\newlineThe maximum profit that the company can earn is $\(8\) \text{ thousand dollars}.

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