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R(q)=-0.31(q-260)^(2)+9,500 A shoe manufacturer determines that its monthly revenue, R(q), in dollars, is given by the function, where q is the number of pairs of shoes sold each month. What is the maximum value of the company's monthly revenue in dollars?

R(q)=−0.31(q−260)2+9,500 R(q)=-0.31(q-260)^{2}+9,500 \newlineA shoe manufacturer determines that its monthly revenue, R(q) R(q) , in dollars, is given by the function, where q q is the number of pairs of shoes sold each month. What is the maximum value of the company's monthly revenue in dollars?

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Full solution

Q. R(q)=−0.31(q−260)2+9,500 R(q)=-0.31(q-260)^{2}+9,500 \newlineA shoe manufacturer determines that its monthly revenue, R(q) R(q) , in dollars, is given by the function, where q q is the number of pairs of shoes sold each month. What is the maximum value of the company's monthly revenue in dollars?
  1. Identify Function Type: Identify the type of function. The revenue function R(q)R(q) is a quadratic function in the form of R(q)=a(q−h)2+kR(q) = a(q - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola.
  2. Determine Parabola Direction: Since the coefficient of (q−260)2(q - 260)^2 is negative (−0.31)(-0.31), the parabola opens downwards, which means the vertex is the maximum point.
  3. Find Vertex Coordinates: The vertex form of the parabola gives us the hh and kk values directly. Here, h=260h = 260 and k=9,500k = 9,500, which means the vertex is at (260,9500)(260, 9500).
  4. Calculate Maximum Revenue: The kk value of the vertex represents the maximum value of the company's monthly revenue. So, the maximum revenue is $9,500.

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