Consider the marginal cost function $$ \mathrm{C}^{\prime}(\mathrm{x})=200+12 \mathrm{x}-0.01 \mathrm{x}^{2} $$ a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units. b. Find the additional cost incurred in dollars when production is increased from 400 units to 450 units.

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To determine the additional cost incurred when increasing production, we need to integrate the marginal cost function $$ \mathrm{C}'(x) = 200 + 12x - 0.01x^2 $$ over the specified range of production levels.

### a. Increasing Production from 100 Units to 150 Units

**Step 1: Set Up the Integral**

$$
\text{Additional Cost} = \int_{100}^{150} (200 + 12x - 0.01x^2) \, dx
$$

**Step 2: Compute the Indefinite Integral**

$$
\int (200 + 12x - 0.01x^2) \, dx = 200x + 6x^2 - \frac{0.01}{3}x^3 + C
$$

**Step 3: Evaluate the Definite Integral**

$$
\begin{align*}
\text{Additional Cost} &= \left[200x + 6x^2 - \frac{0.01}{3}x^3\right]_{100}^{150} \
&= \left(200 \times 150 + 6 \times 150^2 - \frac{0.01}{3} \times 150^3\right) \
&\quad - \left(200 \times 100 + 6 \times 100^2 - \frac{0.01}{3} \times 100^3\right) \
&= (30,000 + 135,000 - 11,250) - (20,000 + 60,000 - 3,333.33) \
&= 153,750 - 76,666.67 \
&= 77,083.33 \text{ dollars}
\end{align*}
$$

**Answer for Part a:**  
The additional cost is **\$77,083.33**.

---

### b. Increasing Production from 400 Units to 450 Units

**Step 1: Set Up the Integral**

$$
\text{Additional Cost} = \int_{400}^{450} (200 + 12x - 0.01x^2) \, dx
$$

**Step 2: Compute the Indefinite Integral**

$$
\int (200 + 12x - 0.01x^2) \, dx = 200x + 6x^2 - \frac{0.01}{3}x^3 + C
$$

**Step 3: Evaluate the Definite Integral**

$$
\begin{align*}
\text{Additional Cost} &= \left[200x + 6x^2 - \frac{0.01}{3}x^3\right]_{400}^{450} \
&= \left(200 \times 450 + 6 \times 450^2 - \frac{0.01}{3} \times 450^3\right) \
&\quad - \left(200 \times 400 + 6 \times 400^2 - \frac{0.01}{3} \times 400^3\right) \
&= (90,000 + 1,215,000 - 303,750) - (80,000 + 960,000 - 213,333.33) \
&= 1,001,250 - 826,666.67 \
&= 174,583.33 \text{ dollars}
\end{align*}
$$

**Answer for Part b:**  
The additional cost is **\$174,583.33**.
**Additional Costs:** a. Increasing production from 100 to 150 units costs **\$77,083.33**. b. Increasing production from 400 to 450 units costs **\$174,583.33**.

Consider the marginal cost function
a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.
b. Find the additional cost incurred in dollars when production is increased from 400 units to 450 units.

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Consider the marginal cost function a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units. b. Find the additional cost incurred in dollars when production is increased from 400 units to 450 units.