The total sales of a company​ (in millions of​ dollars) t months from now are given by Upper S left parenthesis t right parenthesis equals 0.05 t cubed plus 0.5 t squared plus 4 t plus 2

The given sales function is:

\[
S(t) = 0.05t^3 + 0.5t^2 + 4t + 2
\]

Where:
- $ S(t) $ represents the total sales (in millions of dollars).
- $ t $ represents the number of months from now.

This is a cubic function that describes the sales over time. You can use this function to calculate the total sales at any month $ t $, analyze trends, and find the rate of change of sales.

### Analysis:
1. **Instantaneous Sales (At a specific month $ t $):**
   Plug in the value of $ t $ to find the total sales at that particular time.

2. **Rate of Change of Sales (Using Derivatives):**
   To find how fast the sales are changing at a specific time $ t $, take the first derivative of $ S(t) $ with respect to $ t $.

   The first derivative is:
   \[
   S'(t) = \frac{d}{dt} \left( 0.05t^3 + 0.5t^2 + 4t + 2 \right)
   \]
   Simplifying:
   \[
   S'(t) = 0.15t^2 + t + 4
   \]
   This represents the rate of change of sales at any given time.

3. **Acceleration of Sales Growth (Second Derivative):**
   The second derivative tells us how the rate of change of sales is itself changing:
   \[
   S''(t) = \frac{d}{dt} \left( 0.15t^2 + t + 4 \right)
   \]
   Simplifying:
   \[
   S''(t) = 0.3t + 1
   \]
   This indicates whether the sales growth is accelerating or decelerating at time $ t $.

Would you like to calculate sales for a specific time, explore the growth rate in detail, or see something else?

### Related Questions:
1. What will the sales be after 6 months? 
2. How fast are the sales growing at month 10? 
3. When will the sales growth rate start increasing significantly?
4. Can we find when the sales will reach a certain value (e.g., $10 million)?
5. What is the effect of the cubic term $ 0.05t^3 $ on long-term sales?

### Tip:
For better analysis, you can use derivatives to predict trends in growth or decline over time!

The total sales of a company (in millions of dollars) t months from now are given by S(t) = 0.05t^3 + 0.5t^2 + 4t + 2.

Analyze the sales function S(t) = 0.05t^3 + 0.5t^2 + 4t + 2 to calculate total sales and predict sales growth. Learn how to use derivatives to find the rate of change and acceleration in sales, suitable for Grades 10-12.

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