Question

Life expectancies in the United States are always rising because of advances in health care, increased education, and other factors. The rate of change (measured at the end of each year) of life expectancies for women in the United States between 1970 and 2010 can be modeled by f(t) = 0.0004t2 − 0.022t + 0.36 years per year where t is the number of years since 1970. Note: f is measured in years (life expectancy) per year (chronological)Between 1970 and 2010, when was the life expectancy for women growing least rapidly?

Answer 1

The life expectancy for women was growing least rapidly in the year 1997.

Step-by-step explanation:

The rate of change of life expectancies for women in the United States between 1970 and 2010 can be modeled by the function f(t) = 0.0004t^2 - 0.022t + 0.36, where t is the number of years since 1970. To find when the life expectancy for women was growing least rapidly, we need to determine the minimum point on the graph of f(t). This can be found by taking the derivative of f(t) and setting it equal to zero. We then solve for t to find the year.

Taking the derivative of f(t), we get f'(t) = 0.0008t - 0.022. Setting this equal to zero, we have 0.0008t - 0.022 = 0. Solving for t, we find t = 27.5. Since t represents the number of years since 1970, we add 1970 to t and find that the life expectancy for women was growing least rapidly in the year 1997.

Answer 2

1970

Step-by-step explanation:

The function that models the life expectancy for women is the parabola

`f(t)=0.0004t^2-0.022t+0.36 \; (1970\leq t\leq 2010)`

Recall that the speed of growing of a differentiable function at a point t, is given by the derivative at that point.

The derivative of f is

`f'(t)=0.0008t-0.022\; (1970\leq t\leq 2010)`

As we can see, f' represents the equation of a line with a positive slope, so its minimum is reached at the left end of the interval where it is defined, in this case, 1970.

So, the life expectancy for women was growing least rapidly in 1970.