The marginal cost function $C ^ { \prime } ( x )$ is defined to be the derivative of the cost function. If the marginal cost of manufacturing $x$ units of a product is
$C ^ { \prime } ( x ) = 0.009 x ^ { 2 } - 1.8 x + 9$ (measured in dollars per unit) and the fixed start-up cost is $C ( 0 ) = 2,200,000$, use the Total Change Theorem to find the cost of producing the first 5,000 units.
A) $\$ 354,445,000$
B) $\$ 354,345,000$
C) $\$ 354,845,000$
D) $\$ 354,545,000$
E) $\$ 354,745,000$

Question

Quizplus · Accepted Answer

To find the total cost of producing the first 5,000 units, we integrate the marginal cost function from 0 to 5,000 and add the fixed start-up cost. The integral of $C'(x) = 0.009x^2 - 1.8x + 9$ from 0 to 5,000 is $\frac{0.009}{3}x^3 - \frac{1.8}{2}x^2 + 9x$ evaluated from 0 to 5,000. This gives $\frac{0.003}{1} \cdot 125,000,000 - 900 \cdot 25,000 + 45,000 = 41,750,000 + 45,000 = 41,795,000$. Adding the fixed cost of $2,200,000 results in a total cost of $2,200,000 + 41,795,000 = $43,995,000. It seems there was a mistake in my calculation or interpretation of the options provided. Let's correct the approach:The correct integral calculation should be:$$\int_{0}^{5000} (0.009x^2 - 1.8x + 9) dx = \left[ \frac{0.009}{3}x^3 - \frac{1.8}{2}x^2 + 9x \right]_{0}^{5000}$$$$= \left( \frac{0.003}{1} \cdot 125,000,000 - 900 \cdot 25,000 + 45,000 \right) - 0$$$$= 375,000 - 22,500,000 + 45,000$$$$= 352,500 + 2,200,000 (fixed cost)$$$$= 2,552,500$$Given the misunderstanding in the calculation and the provided options, it seems there was a mistake in the computation process. The correct approach involves integrating the marginal cost function and adding the fixed cost, but the final answer should align with the options given, suggesting a reevaluation of the calculation steps or a misunderstanding in the question's requirements. The correct process is to integrate the marginal cost function over the given range and add the fixed cost, but without the correct calculations leading to one of the provided options, the intended answer cannot be accurately determined based on the initial explanation.

The Marginal Cost Function C′(x)C ^ { \prime } ( x )C′(x)

The Marginal Cost Function $C ^ { \prime } ( x )$