The share of aggregate income held by middle-income households in 1970 was 62%, whereas that held by upper-income households was 29%. the corresponding figures in 2014 were 43% and 49%, respectively. the models describing the fall and the rise in the share of the aggregate incomes of these two groups are approximately linear over the period under consideration.† (a) find the mathematical models describing the percent share of aggregate income held by each group from 1970 through 2014. (let t denote the time, in years, with t = 0 corresponding to the beginning of 1970.) middle-income households y = incorrect: your answer is incorrect. upper-income households y = incorrect: your answer is incorrect. (b) find the time when the aggregate income held by upper-income households first exceeded that held by middle-income households. (give the calendar year in which the change occurred.)

  a) y = -19/44t +62; y = 5/11t +29

  b) 2007

Step-by-step explanation:

(a) The two-point form of the equation of a line is useful when two data points are given.

  y = (y2 -y1)/(x2 -x1)(x -x1) +y1

Middle Income

The two given points are (0, 62), (44, 43), so the equation is ...

  y = (43 -62)/(44 -0)(t -0) +62

  y = -19/44t +62

__

Upper Income

The two given points are (0, 29), (44, 49), so the equation is ...

  y = (49 -29)/(44 -0)(t -0) +29

  y = 20/44t +29

  y = 5/11t +29

__

(b) The year in which the shares are equal is found by setting the y-values equal:

  -19/44t +62 = 5/11t +29

  33 = 39/44t

  t = (33)(44)/39 ≈ 37.2

The upper income share exceeded the middle income share in the year 1970 +37.2 = 2007.

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step-by-step explanation:

in problems involving proportions, we can use cross products to test whether two ratios are equal and form a proportion. to find the cross products of a proportion, we multiply the outer terms, called the extremes, and the middle terms, called the means.

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