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The table in the image provides the profit \( P \) (in millions of dollars) for a fast food chain over different years \( t \), where \( t \) is the number of years since January 1, 2007.

The data from the table is as follows:

| Year \( t \) | Profit \( P \) (millions) |
|--------------|----------------------------|
| 0            | -10                        |
| 1            | -14                        |
| 2            | -19                        |
| 3            | -25                        |
| 4            | ?                          |
| 5            | ?                          |

Let's go through the parts of the problem step-by-step:

### a) Find the average rate of change in profit over the one-year intervals from:
1. **2007 to 2008**: This corresponds to \( t = 0 \) to \( t = 1 \).
   \[
   \text{Rate of change} = \frac{P(1) - P(0)}{1 - 0} = \frac{-14 - (-10)}{1} = \frac{-14 + 10}{1} = -4 \text{ million dollars per year}
   \]
2. **2008 to 2009**: This corresponds to \( t = 1 \) to \( t = 2 \).
   \[
   \text{Rate of change} = \frac{P(2) - P(1)}{2 - 1} = \frac{-19 - (-14)}{1} = \frac{-19 + 14}{1} = -5 \text{ million dollars per year}
   \]
3. **2009 to 2010**: This corresponds to \( t = 2 \) to \( t = 3 \).
   \[
   \text{Rate of change} = \frac{P(3) - P(2)}{3 - 2} = \frac{-25 - (-19)}{1} = \frac{-25 + 19}{1} = -6 \text{ million dollars per year}
   \]

### b) Assuming the pattern repeats, find the average rate of change over the one-year intervals from 2010 to 2011 and from 2011 to 2012.

We observe that the rate of change in profit decreases by \( 1 \) million dollars each year. The rates are \( -4, -5, -6 \), so we can assume the next rate of change will be:
1. **2010 to 2011 (from \( t = 3 \) to \( t = 4 \))**:
   \[
   \text{Rate of change} = -7 \text{ million dollars per year}
   \]
   Therefore, the profit for \( t = 4 \) is:
   \[
   P(4) = P(3) + \text{Rate of change} = -25 + (-7) = -32 \text{ million dollars}
   \]
2. **2011 to 2012 (from \( t = 4 \) to \( t = 5 \))**:
   \[
   \text{Rate of change} = -8 \text{ million dollars per year}
   \]
   Therefore, the profit for \( t = 5 \) is:
   \[
   P(5) = P(4) + \text{Rate of change} = -32 + (-8) = -40 \text{ million dollars}
   \]

### c) Study the table of values. What can be inferred about the function \( P(t) \) from \( t = 0 \) to \( t = 5 \)?

From the table and the computed values, we observe that the profit decreases each year and the rate of decrease accelerates by 1 million dollars every year. This suggests that the profit function \( P(t) \) is likely **non-linear** and could possibly be modeled as a quadratic function, since the rate of change is increasing by a constant value.

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Would you like further details or explanations on any part of this solution?

Here are some related questions to deepen understanding:
1. How would you compute the average rate of change for a time interval longer than one year?
2. Can you formulate a quadratic equation for \( P(t) \) based on the given data points?
3. How does the concept of average rate of change relate to the derivative in calculus?
4. What would the graph of this profit function look like over time?
5. How could external factors (e.g., market conditions) impact the trend in profit?

**Tip:** The average rate of change provides a simplified snapshot of how a quantity changes over time, but if the change is accelerating or decelerating, a more detailed model (like quadratic or higher-order polynomials) is often required.

The table gives the profit, P of a fast food chain, where P is in millions of dollars and t is in years since January 1, 2007. a) Find the average rate of change in profit over the one-year intervals from 2007 to 2008, from 2008 to 2009 and from 2009 to 2010. b) Assuming the pattern repeats, find the average rate of change over the one-year intervals from 2010 to 2011 and from 2011 to 2012. c) Study the table of values. What can be inferred about the function P(t) from t = 0 to t = 5?

Learn how to find the average rate of change in profit for a fast food chain over specific time intervals and analyze the trend using a quadratic pattern assumption. Explore calculations from 2007 to 2012, focusing on the behavior of the profit function.

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