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.
You may use either point-slope or slope intercept.
P (t) =  −0.014t+1.94
Correct: Your answer is correct.

(b) Use your model to estimate the population of Philadelphia on July 1, 1984.
Give your numeric answer correct to three decimal places and pick the correct unit from the drop-down menu.

Let's start with the linear model given in part (a):

\[
P(t) = -0.014t + 1.94
\]

This model represents the population \( P(t) \) in millions, where \( t \) is the number of years since July 1, 1970.

For part (b), we need to estimate the population of Philadelphia on July 1, 1984. 

1. First, determine the value of \( t \) corresponding to July 1, 1984. Since \( t = 0 \) represents July 1, 1970, we have:

\[
t = 1984 - 1970 = 14
\]

2. Substitute \( t = 14 \) into the linear model to find \( P(14) \):

\[
P(14) = -0.014(14) + 1.94
\]

3. Calculate the value:

\[
P(14) = -0.196 + 1.94 = 1.744 \text{ million people}
\]

So, the estimated population of Philadelphia on July 1, 1984, is **1.744 million people**.

Would you like more details on this process or any additional explanations? 

Here are 5 related questions you might find interesting:

1. How would the population model change if the decline rate were different?
2. What would the population be in 2020 according to this model?
3. How can we check the accuracy of this linear model?
4. What other models could describe population changes over time?
5. How would the model change if there were a population increase instead of a decrease?

**Tip:** When using linear models, always double-check the slope to ensure it accurately represents the rate of change over time.

Learn how to estimate population changes over time using linear models with a specific example from Philadelphia's population data. This problem involves using linear equations to model population trends and making predictions. Suitable for high school students studying algebra and data analysis.

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