Question

the table lists how poverty level income cutoffs (in dollars) for a family of four have changed over time. use the midpoint formula to approximate the poverty level cutoff in 1987 to the nearest dollar. the poverty level cutoff in 1987 was 12051 dollars. (round to the nearest dollar as needed.)

Explanation:

Step1: Identify the two - year data around 1987

We use the data for 1980 (\(x_1 = 1980,y_1=8552\)) and 1990 (\(x_2 = 1990,y_2 = 13551\)). The mid - point formula for a line segment with endpoints \((x_1,y_1)\) and \((x_2,y_2)\) is \(y-y_1=\frac{y_2 - y_1}{x_2 - x_1}(x - x_1)\). Here \(x = 1987\).
First, calculate the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Substitute \(x_1 = 1980,y_1=8552,x_2 = 1990,y_2 = 13551\) into the slope formula:
\(m=\frac{13551 - 8552}{1990 - 1980}=\frac{4999}{10}=499.9\)

Step2: Use the point - slope form

Using the point - slope form \(y - y_1=m(x - x_1)\) with \(x_1 = 1980,y_1 = 8552,m = 499.9,x = 1987\)
\(y-8552=499.9\times(1987 - 1980)\)
\(y-8552=499.9\times7\)
\(y-8552 = 3499.3\)
\(y=8552+3499.3\)
\(y = 12051.3\)

Answer:

\(12051\)