The share of households in the United States that are earning middle class incom

Question

The share of households in the United States that are earning middle class incomes has been in decline since 1970s. The percentage of households with annual incomes within 50% of the median income is given by P(t)=50.3t^-0.09 (1 t 6) where t measured in decades, with t=1 corresponding to 1980.

a. What percentage of households had annual incomes within 50% of the median income in 2010?

b. How fast was the percentage of households with annual incomes within 50% of the median income decreasing in 2010?

Explanation / Answer

P(t)=50.3t-0.09 (1 t 6)

p(1)= year 1980= 50.3 (1)-0.09=50.3

p(2)= year 1990=50.3 (2)-0.09=47.25

p(3)=year 2000 = 50.3(3)-0.09=45.56

p(4) =year 2010 = 50.3(4)-0.09=44.40

percentage of households with in 50% of median= (44.4/50.3)*100= 88.27%

(b) rate of decrease

dP/dt= 50.3 x (-.09)t(-.09-1)

=-4.57 t-1.09

at 2010 t=4

=-4.57(4)-1.09

=-1.008

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