Final answer:
The minimum unit cost for the copy machines, based on the given quadratic cost function C(x) = 0.2x² – 36x + 20,266, is calculated using the vertex of the parabola. The result, without rounding, is $18,646, representing the lowest cost to produce a machine at an output of 90 machines.
Step-by-step explanation:
The student is asking how to find the minimum unit cost from a quadratic cost function that describes the cost to make each copy machine. The function given is C(x) = 0.2x² – 36x + 20,266. To find the minimum unit cost, we need to find the vertex of the parabola represented by this quadratic equation because the coefficient of x² is positive, indicating a parabola that opens upwards. The vertex form of a quadratic function is C(x) = a(x-h)² + k, where (h, k) is the vertex of the parabola. To find h, we use the formula h = -b/(2a), where a and b are the coefficients from the standard form ax² + bx + c.
Given the equation C(x) = 0.2x² – 36x + 20,266, the coefficients are a = 0.2 and b = -36. Applying the formula h = -b/(2a) yields h = -(-36)/(2 × 0.2) = 36/0.4 = 90. To find the minimum cost, we substitute x = 90 back into the original equation, giving us C(90) = 0.2(90)² – 36(90) + 20,266 = 0.2(8100) – 3240 + 20,266 = 1620 – 3240 + 20,266 = 18,646. This is the minimum unit cost.