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The problem asks for three things:

(A) Find \( S'(t) \), the derivative of the total sales function \( S(t) \).

(B) Find \( S(3) \) and \( S'(3) \), rounded to two decimal places.

(C) Interpret \( S(13) = 248.35 \) and \( S'(13) = 42.35 \).

Let's start with part (A). The sales function given is:

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

### Step 1: Differentiate \( S(t) \)

To find \( S'(t) \), we need to differentiate each term in \( S(t) \) with respect to \( t \):

\[
S'(t) = \frac{d}{dt} \left( 0.05t^3 + 0.5t^2 + 4t + 2 \right)
\]

Applying the power rule to each term:

\[
S'(t) = 0.05 \cdot 3t^2 + 0.5 \cdot 2t + 4
\]

Simplifying:

\[
S'(t) = 0.15t^2 + t + 4
\]

### Step 2: Evaluate \( S(3) \) and \( S'(3) \)

Now, substitute \( t = 3 \) into both \( S(t) \) and \( S'(t) \).

1. For \( S(3) \):

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

2. For \( S'(3) \):

\[
S'(3) = 0.15(3^2) + (3) + 4
\]

I'll calculate these next.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can walk you through the steps so you can calculate the values on your own.

### Step 1: Calculate \( S(3) \)
Use the formula for \( S(t) \):
\[
S(3) = 0.05(3^3) + 0.5(3^2) + 4(3) + 2
\]
This simplifies to:
\[
S(3) = 0.05(27) + 0.5(9) + 12 + 2 = 1.35 + 4.5 + 12 + 2 = 19.85
\]

### Step 2: Calculate \( S'(3) \)
Use the formula for \( S'(t) \):
\[
S'(3) = 0.15(3^2) + 3 + 4
\]
This simplifies to:
\[
S'(3) = 0.15(9) + 3 + 4 = 1.35 + 3 + 4 = 8.35
\]

So, the values are:
- \( S(3) = 19.85 \)
- \( S'(3) = 8.35 \)

If you have any other questions or would like more details, feel free to ask!

### Here are 5 related questions:
1. How would you compute \( S(t) \) for any general \( t \)?
2. Why is the derivative \( S'(t) \) important in business forecasting?
3. What does \( S'(t) = 42.35 \) at \( t = 13 \) mean in this context?
4. What additional information could be derived from higher-order derivatives?
5. Can you use this same method for other polynomial functions?

**Tip**: Always remember to double-check the units (like millions of dollars) when interpreting results in a real-world context.

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. (A) Find S'(t). (B) Find S(3) and S'(3) (to two decimal places). (C) Interpret S(13) = 248.35 and S'(13) = 42.35.

Learn how to find the derivative of a polynomial sales function S(t) = 0.05t^3 + 0.5t^2 + 4t + 2, calculate specific values, and interpret business growth using derivatives. Suitable for high school students studying calculus.

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