Final answer:
The total cost of producing q goods can be found by evaluating a(q). The minimum marginal cost occurs at the value of q that minimizes the derivative of a(q). This value also represents the production level at which the average cost is a minimum. The lowest average cost can be found by substituting this value into a(q). The marginal cost at a specific value of q can be computed using the derivative of a(q).
Step-by-step explanation:
The total cost, C(q), of producing q goods can be found by evaluating the function a(q) = 0.01q² - 0.6q + 13 at the given value of q. For example, to find the total cost of producing 5 goods, substitute q = 5 into the function: a(5) = 0.01(5)² - 0.6(5) + 13 = $7.75.
The minimum marginal cost can be found by finding the derivative of a(q) and solving it for q. The derivative of a(q) is given by a'(q) = 0.02q - 0.6. Set this derivative equal to zero and solve for q: 0.02q - 0.6 = 0 => q = 30. Therefore, the minimum marginal cost occurs at q = 30.
To find the production level at which the average cost is a minimum, we need to find the value of q that minimizes the function a(q). This can be done by finding the critical points of a(q), which are the values of q where the derivative is zero or undefined. In this case, a'(q) = 0.02q - 0.6 = 0 => q = 30. Therefore, the average cost is a minimum at q = 30.
The lowest average cost can be found by substituting the value of q that minimizes the average cost into the function a(q). In this case, a(30) = 0.01(30)² - 0.6(30) + 13 = $4.9. Therefore, the lowest average cost is $4.9.
To compute the marginal cost at q = 30, we can use the derivative of a(q) calculated earlier. The derivative, a'(q) = 0.02q - 0.6, represents the marginal cost function. So, plugging in q = 30 gives us a'(30) = 0.02(30) - 0.6 = $0.