A multiyear analysis was begun in 1987 to determine the perc

A multi-year analysis was begun in 1987 to determine the percentage of American women claiming to be on a weight-loss diet. 7 years into the study, 19\\% of women were found to be on a diet, while 15 years into the study, 7.5\\% of women were found to be on a diet. Use this information to find the equation of the line that models this data. Let \\, y \\, = percentage of women on a diet, and let \\, x \\, = number of years into the study. Give the point-slope form and the slope-intercept form. When entering the point-slope form, keep terms with \\, y \\, on the left and keep terms with \\, x \\, on the right. Decimals are allowed throughout this problem.

Use your equation to predict the percentage of American women on a diet in 2007.
Simplify your answer. The % symbol is already provided, so you do not need to enter it.

The percentage of American women on a diet in 2007 is

Solution

The point-slope form of a linear equation:

(y₂ - y₁) = m(x₂ - x₁)

point (x₁,y₁) = (7,19)

point (x₂,y₂) = (15,7.5)

slope (m) = (y₂ - y₁) / (x₂ - x₁)

m = (7.5 - 19) / (15 - 7)

m = -11.5/8

m = -1.4375

Using slope (m) and point (x₁,y₁):

  y - 19 = -1.4375(x - 7) [point-slope answer]

The slope-intercept form of a linear equation:

y = mx + b

Using point (x₁,y₁) and slope (m) to calculate

y-intercept (b):   

19 = -1.4375(7) + b

19 = -10.0625 + b

b = 19+10.0625

b = 29.0625

Using slope (m) and y-intercept (b):

  y = -1.4375x +29.0625 [slope-intercept answer]

The percentage of women on diet (y) in 2007:

x = 2007 - 1987

x = 20 years into study

y = -1.4375x + 29.0625 (slope-intercept equation)

y =-1.4375(20) + 29.0625

y = 0.3125

  y = 0.3125% of women on diet in 2007