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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 P(x)=-3(x-5)^2+12 What sock price should the company set to earn a maximum profit?

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Full solution

Q. 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 P(x)=-3(x-5)^2+12 What sock price should the company set to earn a maximum profit?
  1. Identify profit function: Identify the profit function and recognize that it is a quadratic function in the form P(x)=−3(x−5)2+12P(x) = -3(x - 5)^2 + 12, which represents a parabola opening downwards.
  2. Find maximum profit: Understand that the maximum profit for a parabola that opens downwards occurs at the vertex of the parabola.
  3. Calculate vertex of parabola: Calculate the vertex of the parabola. Since the profit function is in vertex form P(x)=a(x−h)2+kP(x) = a(x - h)^2 + k, where (h,k)(h, k) is the vertex, we can see that h=5h = 5 and k=12k = 12.
  4. Determine optimal price: Conclude that the maximum profit occurs when x=5x = 5, which means the company should set the price of a pair of socks to $5\$5 to earn the maximum profit.

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