In 1991, the life expectancy of males in a certain country was 72.6 years. In 1996, it was 74.7 years. Let E represent the life expectancy in year t and represent the number of years since 1991.
The linear function E(t) that fits that data is, E(t)=
e = expectancy
t = year - 1991
points are (t,e) (0,72.6) (5, 74.7)
slope = (74.7-72.6) /(5-0) = .42
e(t) = .42 t + b
72.6 = 0 + b so b = 72.6
e(t) = .42 t + 72.6
Now let's see, 2014 - 1991 = 23
so
e(23) = .42(23) + 72.6
e(23) = 82.3
well, that is encouraging :)
To find the linear function that fits the given data, we need to determine the equation of a line in the slope-intercept form: y = mx + b.
Let's assign the following variables:
- E(t): Life expectancy in year t
- t: Number of years since 1991
Given data:
- In 1991: t = 0, E(t) = 72.6
- In 1996: t = 5, E(t) = 74.7
Using the information from the given data, we can find the slope (m) and the y-intercept (b) of the linear function.
Step 1: Finding the slope (m)
The slope is calculated using the formula: m = (change in y) / (change in x)
change in y = E(t2) - E(t1) = 74.7 - 72.6 = 2.1
change in x = t2 - t1 = 5 - 0 = 5
m = (2.1) / (5) = 0.42
Step 2: Finding the y-intercept (b)
The y-intercept can be determined by substituting the values of a known point (t, E(t)) and the slope (m) into the equation y = mx + b. We can use the first given data point, which is (0, 72.6).
Using the point (0, 72.6), we have:
72.6 = (0.42)(0) + b
72.6 = b
Step 3: Writing the linear function E(t)
Now that we have determined the slope (m = 0.42) and the y-intercept (b = 72.6), we can write the linear function E(t) in the slope-intercept form.
E(t) = 0.42t + 72.6
Therefore, the linear function that fits the given data is:
E(t) = 0.42t + 72.6