Question

The cost function for a particular product is given by C(x)=0.001 x³-0.15 x² +12.1 x+110 dollars. where ( 0 ≤ x ≤ 60 ). Find the minimum marginal cost of the product, rounded to the nearest cent.

Answer 1

The minimum marginal cost of the product is $18.10.

Step-by-step explanation:

Marginal Cost Calculation

The marginal cost can be calculated by finding the derivative of the cost function with respect to x. So, we differentiate C(x)=0.001 x³-0.15 x² +12.1 x+110 with respect to x.

C'(x) = 0.003 x² - 0.3 x + 12.1

Now, we set C'(x) equal to zero and solve for x to find the critical points.

0.003 x² - 0.3 x + 12.1 = 0

Using the quadratic formula, x = 30 and x = 133.33

Since x must be between 0 and 60, the critical point x = 133.33 is ignored.

Therefore, the minimum marginal cost occurs at x = 30.

Rounded to the Nearest Cent

To round the minimum marginal cost to the nearest cent, we evaluate C'(30).

C'(30) = 0.003(30)² - 0.3(30) + 12.1 = 18.1

Rounded to the nearest cent, the minimum marginal cost is $18.10.