To minimize the average cost per unit c, we first start with the given equation for total cost:
c = 0.001x3 + 5x + 250
The average cost per unit (AC) is calculated by dividing the total cost by the number of units produced:
AC = c/x = (0.001x3 + 5x + 250)/x = 0.001x2 + 5 + 250/x
To find the value of x that minimizes the average cost, we need to differentiate this average cost function with respect to x and set the derivative to zero:
d(AC)/dx = 0.002x – 250/x2
Setting the derivative equal to zero gives:
0.002x – 250/x2 = 0
Rearranging the equation leads to:
0.002x3 = 250
Now, solve for x:
x3 = 250 / 0.002
x3 = 125000
Taking the cube root of both sides, we find:
x = 50
Thus, the number of units x that produces the minimum average cost per unit is 50 units.