Questions: In 1981 the population density of a country was 59 people per square mile, and in 2001 it was 75 people per square mile. Use a linear function to estimate when the population density reached 89 people per square mile. The population density will reach 89 people per square mile in the year (Round to the nearest whole number as needed.)

In 1981 the population density of a country was 59 people per square mile, and in 2001 it was 75 people per square mile. Use a linear function to estimate when the population density reached 89 people per square mile.

The population density will reach 89 people per square mile in the year (Round to the nearest whole number as needed.)

Transcript text: In 1981 the population density of a country was 59 people per square mile, and in 2001 it was 75 people per square mile. Use a linear function to estimate when the population density reached 89 people per square mile. The population density will reach 89 people per square mile in the year $\square$ (Round to the nearest whole number as needed.)

Solution

Solution Steps

To solve this problem, we can use a linear function to model the change in population density over time. We have two data points: (1981, 59) and (2001, 75). First, we calculate the slope of the line, which represents the rate of change in population density per year. Then, we use the point-slope form of a linear equation to find the year when the population density reaches 89 people per square mile.

Step 1: Determine the Slope of the Linear Function

To find the slope of the linear function that models the population density over time, we use the formula for the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the given data points $(1981, 59)$ and $(2001, 75)$:

\[ m = \frac{75 - 59}{2001 - 1981} = \frac{16}{20} = 0.8 \]

Step 2: Use the Point-Slope Form to Find the Year

We use the point-slope form of a linear equation to find the year when the population density reaches 89 people per square mile. The point-slope form is given by:

\[ y - y_1 = m(x - x_1) \]

Substituting $y = 89$, $y_1 = 59$, $m = 0.8$, and $x_1 = 1981$:

\[ 89 - 59 = 0.8(x - 1981) \]

Solving for $x$:

\[ 30 = 0.8(x - 1981) \]

\[ x - 1981 = \frac{30}{0.8} = 37.5 \]

\[ x = 1981 + 37.5 = 2018.5 \]

Step 3: Round to the Nearest Whole Number

Since we need to round to the nearest whole number, the year when the population density reaches 89 people per square mile is:

\[ x = \text{round}(2018.5) = 2019 \]