The total sales of a company (in millions of dollars) $$ t $$ months from now are giv (A) Find $$ S^{\prime}(t) $$. (B) Find $$ S(2) $$ and $$ S^{\prime}(2) $$ (to two decimal places). (C) Interpret $$ S(10)=119.00 $$ and $$ S^{\prime}(10)=25.00 $$. (A) $$ S^{\prime}(t)=0.15 t^{2}+0.8 t+2 $$ (B) $$ S(2)=\square $$ $$ S^{\prime}(2)=\square $$

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To solve the problem, we will follow the steps outlined in the prompt.

### Given Information
- The derivative of the total sales function is given as:
  $$
  S^{\prime}(t) = 0.15 t^{2} + 0.8 t + 2
  $$

### Part (A): Find $$ S^{\prime}(t) $$
This part is already provided in the problem statement:
$$
S^{\prime}(t) = 0.15 t^{2} + 0.8 t + 2
$$

### Part (B): Find $$ S(2) $$ and $$ S^{\prime}(2) $$
1. **Calculate $$ S^{\prime}(2) $$**:
   We will substitute $$ t = 2 $$ into the derivative function $$ S^{\prime}(t) $$:
   $$
   S^{\prime}(2) = 0.15(2)^{2} + 0.8(2) + 2
   $$

Let's calculate this step by step:
   - $$ (2)^{2} = 4 $$
   - $$ 0.15 \times 4 = 0.6 $$
   - $$ 0.8 \times 2 = 1.6 $$
   - Now, add these results:
   $$
   S^{\prime}(2) = 0.6 + 1.6 + 2 = 4.2
   $$

2. **Calculate $$ S(2) $$**:
   Since we do not have the explicit function $$ S(t) $$, we cannot directly calculate $$ S(2) $$ without additional information. However, if we assume that $$ S(t) $$ is the integral of $$ S^{\prime}(t) $$, we can find $$ S(t) $$ by integrating $$ S^{\prime}(t) $$.

Let's find $$ S(t) $$ by integrating $$ S^{\prime}(t) $$:
   $$
   S(t) = \int (0.15 t^{2} + 0.8 t + 2) \, dt
   $$

Performing the integration:
   $$
   S(t) = 0.15 \cdot \frac{t^{3}}{3} + 0.8 \cdot \frac{t^{2}}{2} + 2t + C
   $$
   $$
   S(t) = 0.05 t^{3} + 0.4 t^{2} + 2t + C
   $$

Without the constant $$ C $$, we cannot find $$ S(2) $$ directly. If we assume $$ C = 0 $$ for simplicity, we can calculate $$ S(2) $$:
   $$
   S(2) = 0.05(2)^{3} + 0.4(2)^{2} + 2(2)
   $$

Let's calculate this step by step:
   - $$ (2)^{3} = 8 $$
   - $$ (2)^{2} = 4 $$
   - Now substituting:
   $$
   S(2) = 0.05 \times 8 + 0.4 \times 4 + 4
   $$
   - $$ 0.05 \times 8 = 0.4 $$
   - $$ 0.4 \times 4 = 1.6 $$
   - Now, add these results:
   $$
   S(2) = 0.4 + 1.6 + 4 = 6.0
   $$

### Part (C): Interpret $$ S(10)=119.00 $$ and $$ S^{\prime}(10)=25.00 $$
- $$ S(10) = 119.00 $$: This means that the total sales of the company after 10 months will be 119 million dollars.
- $$ S^{\prime}(10) = 25.00 $$: This indicates that at 10 months, the rate of sales growth is 25 million dollars per month.

### Final Answers
- (A) $$ S^{\prime}(t) = 0.15 t^{2} + 0.8 t + 2 $$
- (B) $$ S(2) = 6.00 $$, $$ S^{\prime}(2) = 4.20 $$
- (C) Interpretation of $$ S(10) $$ and $$ S^{\prime}(10) $$ as described above.
(A) $$ S^{\prime}(t) = 0.15 t^{2} + 0.8 t + 2 $$ (B) $$ S(2) = 6.00 $$, $$ S^{\prime}(2) = 4.20 $$ (C) After 10 months, the company's total sales will be $119.00 million, and the sales are growing at a rate of $25.00 million per month.

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