According to the US Census, the population of Philadelphia, Pennsylvania was 1.94 million on July 1, 1970. By July 1, 2000, the population had decreased to 1.52 million people.

Assuming this trend continues, answer the following.

(a) Denote 
t = 0
 to represent July 1, 1970. Create a linear model for the population 
P
 in millions as a function of the years
t
 since July 1, 1970.

To create a linear model for the population \( P(t) \) in millions as a function of the years \( t \) since July 1, 1970, we follow these steps:

### Step 1: Define the variables and coordinates.
- Let \( t = 0 \) represent the year 1970.
- Let \( P(t) \) be the population in millions at time \( t \).

From the problem:
- In 1970 (\( t = 0 \)), the population was 1.94 million, so \( P(0) = 1.94 \).
- In 2000 (\( t = 30 \), since 2000 is 30 years after 1970), the population was 1.52 million, so \( P(30) = 1.52 \).

### Step 2: Find the slope of the linear function.
The general form of a linear equation is:
\[
P(t) = mt + b
\]
where \( m \) is the slope and \( b \) is the y-intercept (the population at \( t = 0 \)).

The slope \( m \) is calculated using the formula:
\[
m = \frac{\text{Change in population}}{\text{Change in time}} = \frac{P(30) - P(0)}{30 - 0}
\]

Substitute the values:
\[
m = \frac{1.52 - 1.94}{30 - 0} = \frac{-0.42}{30} = -0.014
\]
So, the slope \( m = -0.014 \).

### Step 3: Write the linear equation.
Now, substitute the slope \( m \) and the initial population \( P(0) = 1.94 \) into the equation:
\[
P(t) = -0.014t + 1.94
\]
This is the linear model for the population \( P(t) \) in millions as a function of the years \( t \) since July 1, 1970.

### Final Answer:
The linear model for the population is:
\[
P(t) = -0.014t + 1.94
\]

Let me know if you want further details or have any questions. Here are 5 questions that expand on the information:

1. What is the predicted population in the year 2020 using the linear model?
2. How does the slope of the model relate to the population trend over time?
3. What would be the population in 1975 according to this model?
4. At what year would the population be predicted to reach 1 million, if the trend continues?
5. Can the model be applied indefinitely into the future, and what limitations might it have?

**Tip:** Linear models are most accurate within the range of the given data. Extrapolation far beyond this range can lead to unreliable predictions.

According to the US Census, the population of Philadelphia, Pennsylvania was 1.94 million on July 1, 1970. By July 1, 2000, the population had decreased to 1.52 million people. Assuming this trend continues, answer the following: (a) Denote t = 0 to represent July 1, 1970. Create a linear model for the population P in millions as a function of the years t since July 1, 1970.

Create a linear model to analyze the population decline of Philadelphia from 1970 to 2000. Learn how to calculate the slope and use it in the slope-intercept form to model real-world scenarios. Suitable for students in Grades 9-12.

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