Question

Some scientists believe there is a limit to how long humans can live. One supporting argument is that during the past​ century, life expectancy from age 65 has increased more slowly than life expectancy from​ birth, so eventually these two will be​ equal, at which​ point, according to these​ scientists, life expectancy should increase no further. In​ 1900, life expectancy at birth was 45 ​years, and life expectancy at age 65 was 75 yr. In​ 2010, these figures had risen to 78.7 and 84.5​, respectively. In both​ cases, the increase in life expectancy has been linear. Using these assumptions and the data​ given, find the maximum life expectancy for humans.

Answer 1

Answer:

The maximum life expectancy for humans is approximately 87 years.

Step-by-step explanation:

We have to calculate the point in which both linear functions (Life expectancy from birth and Life expectancy from age 65) intersect, as this is the point in which is estimated to be the maximum life expectancy for humans.

NOTE: to simplify we will consider t=0 to the year 1900, so year 2010 becames t=(2010-1900)=110.

The linear function for Life expectancy from birth can be calculated as:

$t=0\rightarrow y=45\\\\t=110\rightarrow y=78.7\\\\\\m=\dfrac{\Delta y}{\Delta t}=\dfrac{78.7-45}{110-0}=\dfrac{33.7}{110}=0.3064\\\\\\y=0.3064t+45$

The linear function for Life expectancy from age 65 can be calculated as:

$t=0\rightarrow y=75\\\\t=110\rightarrow y=84.5\\\\\\m=\dfrac{\Delta y}{\Delta t}=\dfrac{84.5-75}{110-0}=\dfrac{9.5}{110}=0.0864\\\\\\y=0.0864t+75$

Then, the time t where both functions intersect is:

$0.3064t+45=0.0864t+75\\\\(0.3064-0.0864)t=75-45\\\\0.22t=30\\\\t=30/0.22\\\\t=136.36$

The time t=136.36 corresponds to the year 1900+136.36=2036.36.

Now, we can calculate with any of both functions the maximum life expectancy:

$y=0.0864(136.36)+75\\\\y=11.78+75\\\\y=86.78\approx87$

The maximum life expectancy for humans is approximately 87 years.