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

1. The rate of change in the problem is the increase of $220 per year in the poverty threshold, which represents the slope of a linear function modeling the poverty threshold. 2. If we set x=0 as 1980, the y-intercept for our function is $4,190. 3. The linear function describing the poverty threshold in terms of years after 1980 is y = 220x + 4190. Using this function, the estimated poverty threshold in 2010 is $11,090 and it will take more than 48 years after 1980 for the threshold to pass $15,000 per year. 4. The accuracy of the model in predicting the most recent poverty threshold would require comparing the value predicted by the linear function to the actual value.

Step-by-step explanation:

1. The piece of information in the problem that represents a rate of change is the increase of $220 per year in the poverty threshold. In a linear function modeling the poverty threshold, this rate of change would be represented by the slope of the function.

2. If we decide that x=0 corresponds to 1980, the y-intercept for our function would be $4,190, which is the poverty threshold in 1980.

3. To write a linear function that describes the poverty threshold in dollars in terms of years after 1980, we can use the slope-intercept form of a linear function: y = mx + b. The slope (m) is the rate of change of $220 per year and the y-intercept (b) is $4,190. So, the function would be y = 220x + 4190. Using this function, we can estimate the poverty threshold in 2010 by substituting x = 2010 - 1980 = 30 into the function: y = 220(30) + 4190 = $11,090. To find the year that the poverty threshold will pass $15,000 per year, we can set y = 15,000 and solve for x: 15,000 = 220x + 4190. Solving this equation, we find x > 48. So, it will take more than 48 years after 1980 for the poverty threshold to pass $15,000 per year.

4. The most recent poverty threshold set by the Census Bureau can be found by using the Internet. Discussing the accuracy of the model in predicting that value would require knowing the most recent poverty threshold and comparing it to the value predicted by the linear function.

Answer 2

Increment in property per year ;

Slope of a linear function

Kindly check explanation for the rest of the answers.

Step-by-step explanation:

The rate of change is the increment in property value per year ; $220

Thsi corresponds to the gradient or slope of a linear function

If 1980 = x and x = 0

Recall :

y = bx + c

Where, b = slope = 220

x = year

y = property threshold

c = intercept value

In 1980, x = 0

y - intercept.

Put x = 0 into the equation :

4190 = 220x + c

4190 = 220(0) + c

4190 = c

3.)

The linear function becomes :

y = 220x + 4190

Property threshold in 2010:

x = 2010 - 1980 = 30

y = 220(30) + 4190

y = 6600 + 4190

y = 10,790

Property threshold in 2010

Year it will exceed 15000

15000 = 220x + 4190

15000 - 4190 = 220x

10810 = 220x

x = 10810 / 220

x = 49.136

That is, property threshold will exceed 15000 after 50 years

1980 + 50 = 2030

Year 2030