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!