The total sales of a company (in millions of dollars) $$ t $$ months from now are given by $$ S(t)=0.01 t^{3}+0.3 t^{2}+3 t+7 $$. (A) Find $$ S^{\prime}(t) $$. (B) Find $$ S^{\prime}(2) $$ and $$ S^{\prime}(2) $$ (to two decimal places). (C) Interpret $$ S(9)=65.59 $$ and $$ S^{\prime}(9)=10.83 $$. (A) $$ S^{\prime}(t)=\square $$

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To find the derivative $$ S^{\prime}(t) $$ of the function $$ S(t) = 0.01 t^{3} + 0.3 t^{2} + 3 t + 7 $$, we will differentiate each term with respect to $$ t $$.
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
  $$\frac{d}{dt}\left(0.01t^{3}+0.3t^{2}+3t+7ight)$$
- step1: Use differentiation rules:
  $$\frac{d}{dt}\left(0.01t^{3}ight)+\frac{d}{dt}\left(0.3t^{2}ight)+\frac{d}{dt}\left(3tight)+\frac{d}{dt}\left(7ight)$$
- step2: Calculate:
  $$0.03t^{2}+0.6t+3+0$$
- step3: Remove 0:
  $$0.03t^{2}+0.6t+3$$
The derivative of the function is given by $$ S^{\prime}(t) = 0.03 t^{2} + 0.6 t + 3 $$.
Now, we will substitute $$ t = 2 $$ into the derivative $$ S^{\prime}(t) $$ to find $$ S^{\prime}(2) $$.
Evaluate the expression by following steps:
- step0: Evaluate:
  $$0.03t^{2}+0.6t+3$$
- step1: Substitute:
  $$0.03	imes 2^{2}+0.6	imes 2+3$$
- step2: Multiply the numbers:
  $$0.12+0.6	imes 2+3$$
- step3: Multiply the numbers:
  $$0.12+1.2+3$$
- step4: Add the numbers:
  $$4.32$$
The value of $$ S^{\prime}(2) $$ is approximately $$ 4.32 $$ (to two decimal places).
Given $$ S(9) = 65.59 $$, this means that the total sales of the company after 9 months will be approximately 65.59 million dollars. The derivative $$ S^{\prime}(9) = 10.83 $$ indicates that the sales are increasing at a rate of approximately 10.83 million dollars per month at that time.
Thus, the answers are: (A) $$ S^{\prime}(t) = 0.03 t^{2} + 0.6 t + 3 $$ (B) $$ S^{\prime}(2) \approx 4.32 $$ (C) $$ S(9) = 65.59 $$ million dollars and $$ S^{\prime}(9) = 10.83 $$ million dollars per month.

The total sales of a company (in millions of dollars) months from now are given by .
(A) Find .
(B) Find and (to two decimal places).
© Interpret and .
(A)

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The total sales of a company (in millions of dollars) months from now are given by . (A) Find . (B) Find and (to two decimal places). © Interpret and . (A)