The marginal cost for producing x units of a certain product is given by the formula MC = 0.001x^2-0.5x+66.5.
A. What is the increase in the cost of the production if the production level is raised from 200 to 400 units?
B. What is the average cost per item of producing 330 items.
MC is the rate of change of the cost, C. So,
C(x) = ∫0.001x^2-0.5x+66.5 dx
(a) ∫[200,400] 0.001x^2-0.5x+66.5 dx = 1966.67
(b) (C(330)-C(0))/330
To find the increase in the cost of production when the production level is raised from 200 to 400 units (question A), we need to calculate the difference in the cost of producing 400 units and 200 units.
1. Calculate the cost at 200 units:
Let's plug in x = 200 into the marginal cost formula:
MC = 0.001(200)^2 - 0.5(200) + 66.5
MC = 0.001(40000) - 100 + 66.5
MC = 40 - 100 + 66.5
MC = 6.5
2. Calculate the cost at 400 units:
Similarly, plug in x = 400 into the marginal cost formula:
MC = 0.001(400)^2 - 0.5(400) + 66.5
MC = 0.001(160000) - 200 + 66.5
MC = 160 - 200 + 66.5
MC = 26.5
3. Calculate the increase in cost:
The increase in the cost of production is the difference between the cost at 400 units and the cost at 200 units:
Increase in cost = MC at 400 units - MC at 200 units
Increase in cost = 26.5 - 6.5
Increase in cost = 20
Therefore, the increase in the cost of production when the production level is raised from 200 to 400 units is 20.
To find the average cost per item of producing 330 items (question B), we need to calculate the total cost of producing 330 units and divide it by the number of units.
1. Calculate the total cost at 330 units:
Let's plug in x = 330 into the marginal cost formula:
MC = 0.001(330)^2 -0.5(330) + 66.5
MC = 0.001(108900) -165 + 66.5
MC = 108.9 - 165 + 66.5
MC = 10.4
2. Calculate the average cost per item:
Average cost = Total cost / Number of items
Average cost = MC at 330 units / 330
Average cost = 10.4 / 330
Average cost ≈ 0.0315 (rounded to four decimal places)
Therefore, the average cost per item of producing 330 items is approximately 0.0315.