2. Life Expectancy Versus Health Care Spending
One article shows a relationship between per capita health care spending h, in thousands of U.S. dollars per person per year, and life expectancy E,
in years.39 The following formula is adapted from that article:
E=72.23 plus 3.85h minus 0.37h exponent 2.
We should point out that this article includes no
justification for the type of model used.
a. Plot the graph of E versus h. Include spending levels from $1000 to $9000 per person per year (so a horizontal span of 1 to 9).

Here is the plot of life expectancy $E$ versus healthcare spending $h$ in thousands of dollars per person per year. It shows the relationship over a spending range of $1000 to $9000 per person annually.

Would you like to explore more details or have any questions?

Here are 5 questions to consider based on this graph:
1. How does life expectancy change as healthcare spending increases beyond $9,000?
2. What does the negative term $-0.37h^2$ suggest about the relationship between healthcare spending and life expectancy?
3. Can the model be extrapolated to spending levels beyond the given range?
4. How does the peak life expectancy in this model compare to real-world data for various countries?
5. Could other factors, such as socioeconomic status, be influencing the results?

**Tip:** The quadratic term in the model introduces a diminishing return effect, meaning that after a certain point, additional healthcare spending yields smaller increases in life expectancy.

Plot the graph of life expectancy E versus healthcare spending h, where E = 72.23 + 3.85h - 0.37h^2, for spending levels from $1000 to $9000 per person per year.

Explore the relationship between healthcare spending and life expectancy using the quadratic model E = 72.23 + 3.85h - 0.37h^2. This problem includes a plot of life expectancy from spending levels of $1000 to $9000 per person per year, revealing how diminishing returns affect life expectancy as healthcare spending increases. Ideal for students in Grades 10-12.

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