Question

A medical equipment industry manufactures X-ray machines. The unit cost C depends on the number of machines made, given by the function C(x) = 0.8x^2 - 352x + 55,203. What is the minimum unit cost?

a) $55,203
b) $352
c) $0.8
d) $176

Answer 1

To find the minimum unit cost, the vertex of the parabola represented by the quadratic function must be calculated using the formula -b/(2a). Substituting the given values, the minimum unit cost is determined to be $176.

Step-by-step explanation:

The student is asking about finding the minimum unit cost of manufacturing X-ray machines based on a given quadratic cost function, C(x) = 0.8x^2 - 352x + 55,203. To find the minimum cost, we need to complete the square or use calculus to find the vertex of the parabola, which, in a quadratic equation of the form ax^2 + bx + c, represents the minimum (if a is positive) or maximum (if a is negative) value of the function.

To find the minimum cost without calculus, we'll use the formula for the x-coordinate of the vertex, which is -b/(2a). Plugging the values from our cost function we get:

x = -(-352) / (2 * 0.8) = 352 / 1.6 = 220

Now we use this x value to find the minimum cost C(220) = 0.8(220)^2 - 352(220) + 55,203. After calculating, we find that the minimum unit cost is $176, which means the correct answer is (d) $176.