The marginal cost of manufacturing x yards of a certain fabric is
C'(x) = 3 − 0.01x + 0.000012x2
(in dollars per yard). Find the increase in cost if the production level is raised from 2000 yards to 4000 yards.
$186,000
both answers are wrong
correct answer = $170,000
correct answer = $170,000
100%
To find the increase in cost when the production level is raised from 2000 yards to 4000 yards, we need to calculate the total cost at each level and then compare the difference.
First, we integrate the marginal cost function C'(x) to find the total cost function C(x). The integral of C'(x) is given by:
C(x) = ∫ (3 − 0.01x + 0.000012x^2) dx
Integrating each term separately, we get:
C(x) = 3x - 0.005x^2 + 0.000004x^3 + C
where C is the constant of integration.
Next, we evaluate the total cost function at x = 2000 and x = 4000 to find the cost at each level:
C(2000) = 3(2000) - 0.005(2000)^2 + 0.000004(2000)^3 + C
C(2000) = 6000 - 20 + 32 + C
C(2000) = 6012 + C
C(4000) = 3(4000) - 0.005(4000)^2 + 0.000004(4000)^3 + C
C(4000) = 12000 - 80 + 128 + C
C(4000) = 12048 + C
To find the increase in cost, we subtract the cost at the lower level from the cost at the higher level:
Increase in cost = C(4000) - C(2000)
Increase in cost = (12048 + C) - (6012 + C)
Increase in cost = 12048 - 6012
Increase in cost = $6036
Therefore, the increase in cost of production when the level is raised from 2000 yards to 4000 yards is $6036.
58,000
calculate C' = dC/dx at x = 2000
then dC = C' dx
dC = C' (4000-2000)
C' = 3 - 20 + 40 = 23
dC = 23 (2000) = 46,000