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A certain company's main source of income is selling socks. The company's annual profit (in millions of dollars) as a function of the price of a pair of socks (in dollars) is modeled by: P(x)=-3(x-5)^(2)+12 What is the maximum profit that the company can earn? million dollars

A certain company's main source of income is selling socks.\newlineThe company's annual profit (in millions of dollars) as a function of the price of a pair of socks (in dollars) is modeled by:\newlineP(x)=−3(x−5)2+12P(x)=-3(x-5)^{2}+12\newlineWhat is the maximum profit that the company can earn?\newlinemillion dollars

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Q. A certain company's main source of income is selling socks.\newlineThe company's annual profit (in millions of dollars) as a function of the price of a pair of socks (in dollars) is modeled by:\newlineP(x)=−3(x−5)2+12P(x)=-3(x-5)^{2}+12\newlineWhat is the maximum profit that the company can earn?\newlinemillion dollars
  1. Identify profit function: Identify the profit function and the form it is in.\newlineThe profit function is given by P(x)=−3(x−5)2+12P(x) = -3(x - 5)^2 + 12. This is a quadratic function in the form of P(x)=a(x−h)2+kP(x) = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola.
  2. Determine vertex of parabola: Determine the vertex of the parabola.\newlineSince the coefficient of the squared term is negative (−3-3), the parabola opens downwards, which means the vertex represents the maximum point on the graph. The vertex (h,k)(h, k) can be read directly from the equation as (5,12)(5, 12).
  3. Interpret vertex in context: Interpret the vertex in the context of the problem.\newlineThe vertex (5,12) (5, 12) tells us that the maximum profit, which is k k in the vertex form, is $12 \$12 million, and it occurs when the price of a pair of socks is $5 \$5 .
  4. State maximum profit: State the maximum profit the company can earn.\newlineThe maximum profit the company can earn is $12 million\$12\,\text{million}.

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