marginal cost a division of ditton industries manufactures the futura model microwave oven. the daily cost…

Explanation:

Step1: Recall marginal - cost formula

The marginal - cost function $C'(x)$ is the derivative of the cost function $C(x)$. Given $C(x)=0.002x^{3}-0.06x^{2}+120x + 5000$, using the power rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we have $C'(x)=0.006x^{2}-0.12x + 120$.

Step2: Calculate marginal cost at $x = 110$

Substitute $x = 110$ into $C'(x)$: [ \begin{align*} C'(110)&=0.006\times(110)^{2}-0.12\times110 + 120\ &=0.006\times12100-13.2 + 120\ &=72.6-13.2 + 120\ &=179.4 \end{align*} ]

Step3: Calculate marginal cost at $x = 190$

Substitute $x = 190$ into $C'(x)$: [ \begin{align*} C'(190)&=0.006\times(190)^{2}-0.12\times190 + 120\ &=0.006\times36100-22.8+120\ &=216.6-22.8 + 120\ &=313.8 \end{align*} ]

Step4: Calculate marginal cost at $x = 320$

Substitute $x = 320$ into $C'(x)$: [ \begin{align*} C'(320)&=0.006\times(320)^{2}-0.12\times320 + 120\ &=0.006\times102400-38.4 + 120\ &=614.4-38.4 + 120\ &=696 \end{align*} ]

Answer:

When $x = 110$, the marginal cost is $$179.40$. When $x = 190$, the marginal cost is $$313.80$. When $x = 320$, the marginal cost is $$696.00$.