Question

the life expectancy for a human in the united states can be modeled by the function l given by l(x) = 42.53 + 13.86 ln x. l(x) gives the life expectancy, in years, for x decades after the year 1900. based on the model, when is the first time that human life expectancy is predicted to be 80 years? (note: 1 decade is 10 years). a late 1914 b early 2003 c late 2040s d early 2050s

Explanation:

Step1: Set L(x)=80

$80 = 42.53 + 13.86\ln x$

Step2: Isolate ln x term

$13.86\ln x = 80 - 42.53 = 37.47$

Step3: Solve for ln x

$\ln x = \frac{37.47}{13.86} \approx 2.704$

Step4: Convert to exponential form

$x = e^{2.704} \approx 15.03$ decades

Step5: Calculate year

$1900 + 15.03×10 = 2050.3$? No, wait: 15.03 decades is 150.3 years after 1900 → 2050.3? Wait correction: Wait 15.03 decades is 150.3 years → 1900+150.3=2050.3? But option B is 2003. Wait mistake: Wait 15.03 decades is 150.3 years? No, 1 decade=10 years: 15.03 decades=150.3 years → 1900+150.3=2050.3, which is early 2050s? But wait recalculate ln x: 37.47/13.86≈2.704, e^2.704≈15.03, so 15.03 decades=150.3 years → 1900+150.3=2050.3 → option D? Wait no, wait original function: L(x)=42.53+13.86 ln x, x is decades after 1900. Let's check x=10 (2000): L(10)=42.53+13.86 ln10≈42.53+13.86×2.3026≈42.53+31.92≈74.45 <80. x=15 (2050): L(15)=42.53+13.86 ln15≈42.53+13.86×2.708≈42.53+37.53≈80.06, which is ~80. So 15 decades after 1900 is 2050, so early 2050s (option D). Wait earlier mistake: yes, x=15 decades=150 years →1900+150=2050, so early 2050s (D).

Corrected Answer: D. Early 2050s

Step1: Set L(x)=80

$80 = 42.53 + 13.86\ln x$

Step2: Isolate ln x

$13.86\ln x = 37.47$

Step3: Compute ln x

$\ln x ≈ 2.704$

Step4: Find x

$x ≈ e^{2.704} ≈15.03$ decades

Step5: Calculate year

$1900 +15.03×10≈2050.3$, so early 2050s.

Answer:

B. Early 2003