Suppose the total cost function for manufacturing a certain product C(x) is given by the function below, where C (x) is measured in dollars and x represents the number of units produced. Find the level of production that will minimize the average cost. (Round your answer to the nearest whole number.)
C(x)=0.2(0.01x^2+132)
How many units?
C(x) is the total cost for x units, so the average cost is C(x)/x per unit.
A(x) = C(x)/x = 0.2(0.01x + 132/x)
to minimize A(x), find where dA/dx = 0
dA/dx = 0.2(0.01 - 132/x^2)
dA/dx = 0 when x = 20√33 = 114.89 = 115
To find the level of production that will minimize the average cost, we need to find the derivative of the average cost function and set it equal to zero, then solve for x.
First, let's find the average cost function. The average cost (AC) is defined as the total cost divided by the number of units produced:
AC(x) = C(x) / x
In this case, the total cost function is given as C(x) = 0.2(0.01x^2 + 132). Substituting this into the average cost function, we have:
AC(x) = (0.2(0.01x^2 + 132)) / x
Now, let's differentiate the average cost function with respect to x using the quotient rule:
d(AC(x))/dx = (x * d/dx(0.2(0.01x^2 + 132)) - (0.2(0.01x^2 + 132)) * d/dx(x)) / (x^2)
Simplifying:
d(AC(x))/dx = (x * (0.004x) - (0.2(0.01x^2 + 132)) * 1) / (x^2)
d(AC(x))/dx = (0.004x^2 - 0.02x^2 - 26.4) / (x^2)
d(AC(x))/dx = (-0.016x^2 - 26.4) / (x^2)
Now we set the derivative equal to zero and solve for x:
(-0.016x^2 - 26.4) / (x^2) = 0
-0.016x^2 - 26.4 = 0
-0.016x^2 = 26.4
x^2 = 26.4 / -0.016
x^2 = -1650
Since the equation has a negative value on the right side, there is no real solution for x. This means that there is no level of production that will minimize the average cost.