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The problem asks to analyze the profit function \( P(s) = -0.30s^2 + 3.5s + 11.15 \), where \( s \) is the number of sweatshirts sold (in thousands), and \( P(s) \) is the profit (in thousands of dollars). Let’s break down the solution step by step.

### a) Calculate the average rate of change in profit for the intervals:
The average rate of change (AROC) of a function \( P(s) \) over an interval \( [s_1, s_2] \) is given by:

\[
\text{AROC} = \frac{P(s_2) - P(s_1)}{s_2 - s_1}
\]

We will calculate this for each of the given intervals.

#### i) Interval \( 1 \leq s \leq 2 \)

\[
\text{AROC} = \frac{P(2) - P(1)}{2 - 1}
\]

#### ii) Interval \( 2 \leq s \leq 3 \)

\[
\text{AROC} = \frac{P(3) - P(2)}{3 - 2}
\]

#### iii) Interval \( 3 \leq s \leq 4 \)

\[
\text{AROC} = \frac{P(4) - P(3)}{4 - 3}
\]

#### iv) Interval \( 4 \leq s \leq 5 \)

\[
\text{AROC} = \frac{P(5) - P(4)}{5 - 4}
\]

I'll calculate these now.The average rates of change in profit for the given intervals are:

i) \( 1 \leq s \leq 2 \): AROC = 2.6  
ii) \( 2 \leq s \leq 3 \): AROC = 2.0  
iii) \( 3 \leq s \leq 4 \): AROC = 1.4  
iv) \( 4 \leq s \leq 5 \): AROC = 0.8  

### b) As the number of sweatshirts sold increases, the average rate of change in profit decreases. This suggests that the profit increases at a decreasing rate as more sweatshirts are sold. This means the profit is growing more slowly with each additional sweatshirt sold.

### c) To predict if the rate of change in profit will stay positive, we can observe the function. The quadratic term in \( P(s) \) is negative, indicating that the function is concave down. As a result, the rate of change (slope) will continue to decrease. Eventually, the slope will become zero, and beyond that point, it will become negative, indicating that profit will start decreasing.

Do you want further details on any part? Here are 5 related questions to expand on this topic:

1. How can you determine the exact point where profit stops increasing?
2. What is the significance of the negative quadratic term in \( P(s) \)?
3. How would you interpret the maximum profit in this context?
4. Can you calculate the instantaneous rate of change for this function?
5. How would changes in the coefficients of the quadratic equation affect the profit function?

**Tip**: The rate of change is related to the derivative of the function. The average rate of change gives you the overall trend in an interval, but the instantaneous rate of change can show more specific behavior at any point.

A company that sells sweatshirts finds that the profit can be modeled by P(s) = -0.30s^2 + 3.5s + 11.15, where P(s) is the profit in thousands of dollars, and s is the number of sweatshirts sold (expressed in thousands). Calculate the average rate of change in profit for the intervals and make predictions about future trends.

This problem involves analyzing a company's profit using a quadratic equation model. Calculate the average rate of change in profit for specified intervals and predict future trends. Suitable for high school students exploring algebra and quadratic functions.

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