Key Characteristics of Quadratics

Question

A medical equipment industry manufactures X-ray machines. The unit cost C

 (the cost in dollars to make each X-ray machine) depends on the number of machines made. If x

 machines are made, then the unit cost is given by the function =Cx+−0.8x2320x45,930

. How many machines must be made to minimize the unit cost?



Answer 1

To find the minimum value of the unit cost, we need to find the critical points of the function and determine which is the global minimum. To do this, we'll find the derivative of the function and set it equal to zero, then solve for x.

dC/dx = -0.8x^2 + 45,930

To find the critical points, we set dC/dx = 0 and solve for x:

0 = -0.8x^2 + 45,930 0.8x^2 = 45,930 x^2 = 57,413.75 x = sqrt(57,413.75) = 243.22

The critical point is x = 243.22. To determine if this is a minimum or maximum, we'll use the second derivative test.

The second derivative of the function is:

d^2C/dx^2 = -1.6x



Since the second derivative is negative for all values of x, the critical point is a local minimum. So, to minimize the unit cost, the company must manufacture 243.22 X-ray machines.