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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 \newlineWhat is the maximum profit that the company can earn?

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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 \newlineWhat is the maximum profit that the company can earn?
  1. Identify profit function: Identify the profit function and its form.\newlineThe profit function given is 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: 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 as h=5h = 5 and k=12k = 12.
  3. Calculate maximum profit: Calculate the maximum profit. The maximum profit occurs at the vertex of the parabola. Since the vertex is at (5,12)(5, 12), the maximum profit is \$\(12\) million.
  4. Verify result: Verify the result.\(\newline\)To ensure there are no math errors, we can confirm that the quadratic function is in the correct form and that we have correctly identified the vertex. The function is indeed a downward-opening parabola, and the vertex is correctly found at \((5, 12)\). No further calculations are needed.

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