Question

Help, need to finish this class asap

A medical equipment industry manufacturers 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 C(x)=0.6x^2-372x+63,308. How many machines must be made to minimize the unit cost? Do not round

Answer 1

To find the number of machines that minimize the unit cost, we need to find the value of x that minimizes the function C(x). We can do this by finding the critical points of C(x) and determining which one corresponds to a minimum.

First, we take the derivative of C(x) with respect to x:

C'(x) = 1.2x - 372

Then, we set C'(x) equal to zero and solve for x:

1.2x - 372 = 0

1.2x = 372

x = 310

So the critical point is x = 310.

To determine whether this critical point corresponds to a minimum or a maximum, we need to examine the second derivative of C(x):

C''(x) = 1.2

Since C''(x) is positive for all values of x, we know that the critical point x = 310 corresponds to a minimum.

Therefore, the number of machines that must be made to minimize the unit cost is 310.