Based on data from 1920 and projected to 2050 , the expected life span of people in Country A can be described by the function $f(x)=10.965+14.318 \ln x$, where $x$ is the number of years from 1900 to the person's birth year.
(a) Estimate the birth year for which the expected life span is 75 years.
(b) Use graphical methods to determine the birth year for which the expected life span is 75 years. Does this agree with the solution in part (a)?
(a) The birth year is approximately $\square$ for which the expected life span is 75 years.
(Round to the nearest whole number as needed.)
Answer
Final Answer: The birth year for which the expected life span is 75 years is approximately \(\boxed{1988}\).
Step 1 ：Based on data from 1920 and projected to 2050, the expected life span of people in Country A can be described by the function \(f(x)=10.965+14.318 \ln x\), where \(x\) is the number of years from 1900 to the person's birth year.
Step 2 ：(a) To estimate the birth year for which the expected life span is 75 years, we need to set the function \(f(x)\) equal to 75 and solve for \(x\). This will give us the number of years from 1900 to the person's birth year. Adding this to 1900 will give us the birth year.
Step 3 ：The calculated birth year is a decimal, so we need to round it to the nearest whole number. The birth year is approximately 1987.5615965207198.
Step 4 ：After rounding, the birth year is 1988.
Step 5 ：(b) Using graphical methods to determine the birth year for which the expected life span is 75 years, we find that this agrees with the solution in part (a).
Step 6 ：Final Answer: The birth year for which the expected life span is 75 years is approximately \(\boxed{1988}\).
