Life expectancy in the U.S. is steadily increasing due to medical advancements and the increased awareness of maintaining a healthy lifestyle. The life expectancies for men and women in the U.S. can be modeled by the following functions:
[tex]
\begin{array}{l}
W(x)=0.126 x+76.74 \\
M(x)=0.169 x+69.11
\end{array}
[/tex]
where [tex]$W(x)$[/tex] represents the life expectancy for women and [tex]$M(x)$[/tex] represents the life expectancy for men, and [tex]$x$[/tex] represents the number of years since 1975. ([tex]$x=0$[/tex] corresponds to the year 1975, [tex]$x=5$[/tex] corresponds to the year 1980 and so on.) Write an inequality to determine the birth years of women whose life expectancy is at least 85.
A. [tex]$0.126 x+76.74 \leq 85$[/tex]
B. [tex]$0.126 x+76.74 \geq 85$[/tex]
C. [tex]$0.126 x+76.74\ \textless \ 85$[/tex]
D. [tex]$0.126 x+76.74=85$[/tex]
Answer (2)
The problem gives the life expectancy function for women as W ( x ) = 0.126 x + 76.74 .
We want to find the inequality representing women whose life expectancy is at least 85.
Set up the inequality W ( x ) g e 85 .
Substitute the expression for W ( x ) to get 0.126 x + 76.74 g e 85 .
Explanation
Understanding the Problem The problem states that the life expectancy for women in the U.S. is modeled by the function W ( x ) = 0.126 x + 76.74 , where x is the number of years since 1975. We want to find the inequality that represents the birth years of women whose life expectancy is at least 85.
Setting up the Inequality We are given that the life expectancy W ( x ) must be at least 85. This can be written as W ( x ) g e 85 .
Substituting the Expression Substitute the expression for W ( x ) into the inequality: 0.126 x + 76.74 g e 85
Final Inequality Therefore, the correct inequality is 0.126 x + 76.74 g e 85 .
Examples
Understanding life expectancy trends is crucial in healthcare and urban planning. For instance, if city planners know that women in a certain area are expected to live longer, they can allocate more resources to elderly care facilities and programs. Similarly, healthcare providers can tailor their services to address the specific health needs of an aging female population, ensuring better quality of life and resource management. This proactive approach, guided by mathematical models, helps in creating more sustainable and supportive communities.
To find the birth years of women with a life expectancy of at least 85, we set up the inequality 0.126 x + 76.74 ≥ 85 . The correct option that represents this is option B. This inequality illustrates how life expectancy is expected to change over the years since 1975.
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