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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 sock price should the company set to earn a maximum profit? 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 sock price should the company set to earn a maximum profit?\newlinedollars\text{dollars}

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Full solution

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 sock price should the company set to earn a maximum profit?\newlinedollars\text{dollars}
  1. Identify Function Type: Identify the type of function given for the profit P(x)P(x). The function P(x)=−3(x−5)2+12P(x) = -3(x-5)^2 + 12 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 Parabola Vertex: 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 found directly from the equation P(x)=−3(x−5)2+12P(x) = -3(x-5)^2 + 12, which gives h=5h = 5 and k=12k = 12.
  3. Find Sock Price: Find the sock price for maximum profit.\newlineThe sock price that corresponds to the maximum profit is the x-value of the vertex, which is h=5h = 5 dollars.
  4. Check Result: Check the result.\newlineSince the parabola opens downwards and the vertex is the highest point, the sock price of $5\$5 will indeed give the maximum profit according to the model.

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