Question

The United States Census Bureau uses demographic information to set a poverty threshold that is used in to determine how many Americans are living in poverty based on annual income. For an individual on her own, the poverty threshold was $4,190 in 1980 and has increased by about $220 per year since then.

1. Which piece of information in the problem is a rate of change? What would that represent in a linear function modeling the poverty threshold?

2. When modeling information that changes with time, we almost never use the actual time--whether it's clock time or year--as input. Instead, we chose a beginning time for the problem and call that x=0. In this case, we would decide that x=0 corresponds to 1980 since that's the earliest time we have information for. In that case, what is the y-intercept for our function?

3. Write a linear function that describes the poverty threshold in dollars in terms of years after 1980. Then use your function to estimate the poverty threshold in 2010, and the year that it will pass $15,000 per year.

4. Use the Internet to find the most recent poverty threshold as set by the census bureau, and discuss how accurately your model predicted that value.

Answer 1

Q1. The rate of change for the poverty threshold per year is $220.

Which represents the slope in the linear model.

Q2. Considering the year 1980 as x=0 the poverty threshold at that year,

which is $4190 is the y-intercept in the model.

Q3. The linear model can be given by

y = 4190 + 220*x

The estimate of the poverty threshold in 2010 that is when x =30 is

y = 4190 +220 *30

= 10790

The estimate of the poverty threshold in 2010 is $10,790

The year in which the Poverty threshold will pass $15,000 per year is

y = 4190 +220*x

15000-4190 =220*x

10810/220 = x

that is x= 49.13 =49 year.

Hence,  the Poverty threshold will pass $15,000 per year at 49 years after 1980 that is in 2029.