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.
marginal cost is the derivative of cost with x. the cost of one more at number x
so
integrate your function
C(x) = (.001/3)x^3 - (0.5/2)x^2 + 66.5 x + k
assume when x = 0, c = 0 so k = 0 (not really true because you have to set up to build the first one but you need that for part B)
at x = 400
C(400)=(.001/3)(400)^3 - (0.5/2)400^2 + 66.5 (400)
C(300) = same with 300
subtract
average = total C(330)/330
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.
A. To find the increase in the cost of production, we need to calculate the difference in cost between producing 400 units and producing 200 units.
First, let's calculate the cost for producing 200 units:
MC = 0.001x^2 - 0.5x + 66.5
MC(200) = 0.001(200)^2 - 0.5(200) + 66.5
MC(200) = 0.001(40000) - 100 + 66.5
MC(200) = 40 - 100 + 66.5
MC(200) = 6.5
Next, let's calculate the cost for producing 400 units:
MC = 0.001x^2 - 0.5x + 66.5
MC(400) = 0.001(400)^2 - 0.5(400) + 66.5
MC(400) = 0.001(160000) - 200 + 66.5
MC(400) = 160 - 200 + 66.5
MC(400) = 26.5
To find the increase in cost, we subtract the cost of producing 200 units from the cost of producing 400 units:
Increase in cost = MC(400) - MC(200)
Increase in cost = 26.5 - 6.5
Increase in cost = $20
Therefore, the increase in the cost of production is $20.
A. To find the increase in the cost of production when the production level is raised from 200 to 400 units, we need to calculate the marginal cost for both levels and then subtract the initial cost from the final cost.
First, let's calculate the marginal cost at 200 units:
MC = 0.001x^2 - 0.5x + 66.5
MC(200) = 0.001(200)^2 - 0.5(200) + 66.5
MC(200) = 0.001(40000) - 100 + 66.5
MC(200) = 40 - 100 + 66.5
MC(200) = 6.5
Next, let's calculate the marginal cost at 400 units:
MC = 0.001x^2 - 0.5x + 66.5
MC(400) = 0.001(400)^2 - 0.5(400) + 66.5
MC(400) = 0.001(160000) - 200 + 66.5
MC(400) = 160 - 200 + 66.5
MC(400) = 26.5
Now, we can find the increase in cost:
Increase in cost = MC(400) - MC(200)
Increase in cost = 26.5 - 6.5
Increase in cost = 20
Therefore, the cost of production increases by 20 when the production level is raised from 200 to 400 units.
B. To find the average cost per item of producing 330 items, we need to divide the total cost by the number of items.
First, let's calculate the total cost of producing 330 items:
MC = 0.001x^2 - 0.5x + 66.5
Total cost = MC(330) * 330
Total cost = (0.001(330)^2 - 0.5(330) + 66.5) * 330
Total cost = (0.001(108900) - 165 + 66.5) * 330
Total cost = (108.9 - 165 + 66.5) * 330
Total cost = 10.4 * 330
Total cost = 3432
Now, we can find the average cost per item:
Average cost per item = Total cost / Number of items
Average cost per item = 3432 / 330
Average cost per item ≈ 10.4
Therefore, the average cost per item of producing 330 items is approximately 10.4.