Find linear models for each set of data. In what year will the two quantities be equal?
Year-1970 1975 1980 1985 1990 1995 2000
men (years) 67.6 69.3 70.4 71.6 72.2 72.8 74.9
women (years) 75.1 76.8 77.5 78.2 78.8 79.2 79.9
Let x be the number of years since 1970. What is the linear model for men?
Let x be the number of years since 1970. What is the linear model for women?
At what year would they be the same?
To find the linear models for the data, we can use the formula for a linear equation, y = mx + b, where y is the dependent variable (in this case, the age) and x is the independent variable (in this case, the number of years since 1970).
For the men's data:
Let y be the age of men and x be the number of years since 1970.
We can find the slope (m) using two points from the data: (0, 67.6) and (30, 74.9).
m = (74.9 - 67.6) / (30 - 0) = 7.3 / 30 = 0.2433 (approx.)
To find the y-intercept (b), we can substitute one of the points into the equation:
67.6 = 0.2433(0) + b
b = 67.6
Therefore, the linear model for men's age is y = 0.2433x + 67.6.
For the women's data:
Let y be the age of women and x be the number of years since 1970.
We can find the slope (m) using two points from the data: (0, 75.1) and (30, 79.9).
m = (79.9 - 75.1) / (30 - 0) = 4.8 / 30 = 0.16 (approx.)
To find the y-intercept (b), we can substitute one of the points into the equation:
75.1 = 0.16(0) + b
b = 75.1
Therefore, the linear model for women's age is y = 0.16x + 75.1.
To find the year when the two quantities will be equal, we can set the two equations equal to each other and solve for x:
0.2433x + 67.6 = 0.16x + 75.1
0.0833x = 7.5
x ≈ 90.09
Since x represents the number of years since 1970, we can find the year by adding 1970 to x:
1970 + 90.09 ≈ 2060
Therefore, the two quantities (men and women's ages) will be equal in the year 2060.