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?
round 3 decimal places
To find the linear model for the data, we can use the formula for the equation of a line in slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
First, let's find the linear model for men:
Using the given data points (1970, 67.6) and (2000, 74.9), we can find the slope:
m = (74.9 - 67.6) / (2000 - 1970) = 0.73
To determine the y-intercept, we can plug one of the points into the equation and solve for b:
67.6 = 0.73*1970 + b
67.6 = 1438.1 + b
b = 67.6 - 1438.1
b ≈ -1370.5
Therefore, the linear model for men is: men = 0.73x - 1370.5
Now, let's find the linear model for women:
Using the given data points (1970, 75.1) and (2000, 79.9), we can find the slope:
m = (79.9 - 75.1) / (2000 - 1970) = 0.76
To determine the y-intercept, we can plug one of the points into the equation and solve for b:
75.1 = 0.76*1970 + b
75.1 = 1497.2 + b
b = 75.1 - 1497.2
b ≈ -1422.1
Therefore, the linear model for women is: women = 0.76x - 1422.1
To find the year when the two quantities will be equal, we set the two linear models equal to each other and solve for x:
0.73x - 1370.5 = 0.76x - 1422.1
Simplifying the equation:
-51.6 = 0.03x
Dividing both sides by 0.03:
x ≈ -1720
Since x represents the number of years since 1970, we can subtract 1970 from x to find the year:
1970 - 1720 = 250
Therefore, the two quantities will be equal in the year 1970 + 250 = 2220.