The marginal cost of manufacturing x yards of a certain fabric is $$C ^ {t } ( x ) = 3 - 0.01 x + 0.000006 x ^ { 2 }$$
in dollars per yard. Find the increase in cost if the production level is raised from 1500 yards to 5500 yards.
A) $\$ 188,000.00$
B) $\$ 198,000.00$
C) $\$ 178,000.00$
D) $\$ 218,000.00$
E) $\$ 208,000.00$

Question

Quizplus · Accepted Answer

To find the increase in cost when production is raised from 1500 yards to 5500 yards, we need to integrate the marginal cost function $C^t(x) = 3 - 0.01x + 0.000006x^2$ from 1500 to 5500. The integral of $C^t(x)$ gives the total cost function, and the difference in total cost between 5500 yards and 1500 yards gives the increase in cost.The integral of $C^t(x)$ is $\int (3 - 0.01x + 0.000006x^2) dx = 3x - 0.005x^2 + 0.000002x^3 + C$, where $C$ is the constant of integration.Evaluating this from 1500 to 5500 gives:$$Total\ Cost\ at\ 5500\ yards - Total\ Cost\ at\ 1500\ yards$$$$= (3(5500) - 0.005(5500)^2 + 0.000002(5500)^3) - (3(1500) - 0.005(1500)^2 + 0.000002(1500)^3)$$$$= 198,000$$Therefore, the increase in cost is $198,000.00.

The Marginal Cost of Manufacturing X Yards of a Certain Ct(x)=3−0.01x+0.000006x2C ^ {t } ( x ) = 3 - 0.01 x + 0.000006 x ^ { 2 }Ct(x)=3−0.01x+0.000006x2

The Marginal Cost of Manufacturing X Yards of a Certain $C ^ {t } ( x ) = 3 - 0.01 x + 0.000006 x ^ { 2 }$