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+133)

? units

Don't know

To find the level of production that will minimize the average cost, we need to first find the average cost function.

The average cost (AC) is given by the total cost (C) divided by the number of units produced (x):

AC(x) = C(x) / x

We can substitute the given total cost function C(x) into the average cost function:

AC(x) = [0.2(0.01x^2 + 133)] / x

Let's simplify this expression further.

AC(x) = (0.02x^2 + 26.6) / x

To find the level of production that will minimize the average cost, we can take the derivative of the average cost function and set it equal to zero:

AC'(x) = (0.02(2x)(x) - (0.02x^2 + 26.6)(1)) / x^2
AC'(x) = (0.04x^2 - 0.02x^2 - 26.6) / x^2
AC'(x) = (0.02x^2 - 26.6) / x^2

Setting AC'(x) equal to zero:

0.02x^2 - 26.6 = 0

Solving for x, we get:

0.02x^2 = 26.6
x^2 = 26.6 / 0.02
x^2 = 1330
x ≈ √1330
x ≈ 36.5

Rounding the answer to the nearest whole number, the level of production that will minimize the average cost is approximately 37 units.

To find the level of production that will minimize the average cost, we need to find the value of x that corresponds to the minimum average cost. Here's how you can do it:

1. The average cost is the total cost divided by the number of units produced, so we can represent it as C_avg(x) = C(x)/x.

2. Substitute C(x) from the given equation: C_avg(x) = (0.2(0.01x^2+133))/x.

3. Simplify the expression for C_avg(x): C_avg(x) = 0.002x + 26.6/x.

4. To find the minimum average cost, we need to find the value of x that makes the derivative of C_avg(x) equal to zero.

5. Differentiate C_avg(x) with respect to x: C_avg'(x) = 0.002 - 26.6/x^2.

6. Set C_avg'(x) equal to 0 and solve for x: 0.002 - 26.6/x^2 = 0.

7. Multiply through by x^2 to eliminate the denominator: 0.002x^2 - 26.6 = 0.

8. Add 26.6 to both sides of the equation: 0.002x^2 = 26.6.

9. Divide both sides of the equation by 0.002: x^2 = 13,300.

10. Take the square root of both sides of the equation: x = sqrt(13,300).

11. Calculate the square root: x ≈ 115.28.

12. Round the answer to the nearest whole number: x ≈ 115.

Therefore, the level of production that will minimize the average cost is approximately 115 units.

average cost = totalcost/number of units

a(x) = C(x)/x
= 0.2(0.01x + 133/x)

minimum avg cost is where a'(x) = 0
a'(x) = .02(.01 - 133/x^2)
x = 10√133 = 115.3, so 115