Question

CENSUS The population of Laredo, Texas, was about 215,500 in 2007. It was about 123,000 in 1990. If we assume that the population growth is constant, write a linear equation with an integer slope to represent p, Laredo’s population t years after 1990.

Answer 1

The linear equation with an integer slope representing Laredo's population growth p(t) after 1990 is p(t) = 5441t + 123,000, with t being the number of years after 1990.

Step-by-step explanation:

To find the linear equation representing the population growth of Laredo, Texas after 1990, we use two known data points: the population in 1990 and the population in 2007. We get t=0 for the year 1990, with a population of 123,000, and t=17 for the year 2007, with a population of 215,500.

The slope of the population growth equation can be calculated using the formula for the slope of a line between two points (y2 - y1) / (x2 - x1), which gives us (215,500 - 123,000) / (17 - 0). This results in a slope of 92,500 / 17, which simplifies to approximately 5,441.18. However, since we are asked for an integer slope, we round this to the nearest integer, 5,441.

Now, using the point-slope form y - y1 = m(x - x1) and substituting the slope (5,441) and the point from 1990 (t=0, p=123,000), we get the linear equation p(t) = 5441t + 123,000, where p represents the population and t is the number of years after 1990.

Answer 2

The linear equation to represent Laredo's population p t years after 1990 is:

p = 123,000 + 2,500t

where:

p is the population of Laredo in year 1990 + t

t is the number of years after 1990

Step-by-step explanation:

We are given that the population in 1990 (p₀) is 123,000 and the population in 2007 (p₁⁷) is 215,500. Since we want a linear equation with an integer slope, we need to find the population growth per year.

Calculate the population growth over 17 years: 215,500 - 123,000 = 92,500

Divide the total population growth by the number of years to find the annual growth rate: 92,500 / 17 years ≈ 5,441 people per year

Since we want an integer slope, round the annual growth rate to the nearest integer: 5,441 ≈ 2,500 people per year

Therefore, the population increases by 2,500 people each year.

Set up the linear equation:

p represents the population at any year after 1990 (p₀ + t years)

123,000 is the initial population in 1990 (p₀)

2,500 is the annual population growth (slope)

t is the number of years after 1990

p = 123,000 + 2,500t

This equation accurately represents Laredo's population growth with an integer slope, making it easy to calculate the population for any year after 1990.