**Example 4****:** Write a mathematical model for the population of this city over the given period of time and use that to predict the population in 2010.

Year | 1970 | 1980 | 1990 | 2000 |

Pop’n (thousands) | 234 | 289.5 | 345 | 400.5 |

**Solution**:

**1. What are the two variables? And what are their units? **

Ans. Year and population. Year is in years and population is in thousands of people.

**2. What will we predict? (Make it y.) What are some points?**

Ans. We will predict population. So y = population. Year is the input variable. However, as we saw Example 3, it will be more convenient to have the input variable x = years since 1970. Then x is greater than or equal to zero. Obviously y must also be greater than or equal to zero.

1970 has pop’n of 234. So x = 0 and y = 234. Use the point (0, 234)

1980 has pop’n of 289.5. So x = 10 and y = 289.5. Use the point (10,289.5)** **

**3. Is a linear model appropriate?**

year | year – 1970 | popn |

1970 | 0 | 234 |

1980 | 10 | 289.5 |

1990 | 20 | 345 |

2000 | 30 | 400.5 |

The graph clearly indicates that a linear model is appropriate for this population growth over this time period.

**4. Slope: Find the slope: m = rise / run**

Let (0,234) be the first point and (10,289.5) be the second point.

[latex]m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}=\frac{289.5-234}{10-0}=\frac{55.5}{10}=5.55[/latex]

**Interpret the slope, using the units of the numbers in the problem:**

(As x increases by 1, y increases by m.)

For each year that goes by, population will increase by 5.55 thousand people.

**5. Write the formula for the linear relationship:**

(Choose either point and use the point-slope form of the line. Then simplify it to the slope-intercept form of the line.)

Choose (0,234) for the point and use the slope of 5.55 that I just computed.

[latex]\begin{align}&y-{{y}_{0}}=m(x-{{x}_{0}})\\&y-234=5.55(x-0)\\&y-234=5.55x\\&y=5.55x+234\\\end{align}[/latex]

**6. Interpret the y-intercept.**

*(The value of y is b when [latex]x=0[/latex].)*

When [latex]x=0[/latex], that is, in 1970, the population is 234 thousand people.

7. Use the formula to make the requested prediction. Write the result in a sentence, with units.

For 2010, [latex]x=2010-1970=40[/latex].

[latex]\begin{align}&y=5.55x+234\\&y=5.55(40)+234\\&y=222+234\\&y=456\\\end{align}[/latex]

So, in 2010, the model predicts that the population will be 456 thousand people.

**8. Sketch a graph from the original problem information, use it to estimate the answers to the prediction questions, and determine whether your answers from the algebraic formula are reasonable.**

We take the same graph as before and extend it to the right so that we can estimate a population value for x = 40.

The value we see from the graph is approximately 450 thousand people. That is consistent with the value we obtained from the formula.

The prediction is reasonable.