Final answer:
The minimum unit cost for manufacturing airplane engines, as given by the function C(x) = 0.9x² - 198x + 28964, is found by calculating the vertex of the quadratic function. The calculation reveals that the minimum cost occurs at x = 110 units and is d. $18074.
Step-by-step explanation:
The student has asked to find the minimum unit cost for manufacturing airplane engines, given the cost function C(x) = 0.9x² - 198x + 28964. To find the minimum cost, we need to find the vertex of the parabola represented by this quadratic function. The vertex form of a quadratic function is given by C(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. Here, the vertex will provide us with the minimum unit cost.
The x-coordinate of the vertex (h) can be found using the formula -b/(2a) for a quadratic equation ax² + bx + c. For the given function, a = 0.9 and b = -198, so h = -(-198) / (2 * 0.9) = 198 / 1.8 = 110. The minimum cost will be at x = 110 units.
Substituting x = 110 into the cost function C(x) gives us the minimum unit cost C(110) = 0.9(110)² - 198(110) + 28964 = 0.9(12100) - 21780 + 28964 = 10890 - 21780 + 28964 = $18074. Therefore, the minimum unit cost is $18074.