Questions: The data on the right has an equation close to L=0.16 t+73.71, where L is the life expectancy at birth (in years) of a person born t years after 1980 (see the table). Complete parts a through d. Year of Birth Life Expectancy (years) - - 1980 73.7 1985 74.7 1990 75.4 1995 75.8 2000 77.0 2005 77.9 2006 78.1 a. Use the linear model to predict the life expectancy of a person born in 2014. The life expectancy of a person born in 2014 will be years. (Type an integer or a decimal.) b. Use the linear model to predict the birth year in which the life expectancy of a person will be 78 A person born in will have a life expectancy of 78.8 years. (Round to the nearest whole number as needed.) c. Use the linear model to estimate what the life expectancy was at 1976. The life expectancy of a person born in 1976 was years. (Type an integer or a decimal.)

Solution Steps

To solve the given problems using the linear model $L = 0.16t + 73.71$, we will follow these steps:

a. For predicting the life expectancy of a person born in 2014, we need to calculate $t$ as the number of years after 1980 and then substitute it into the linear equation.

b. To find the birth year when the life expectancy is 78, we will set $L = 78$ and solve for $t$, then convert $t$ back to the birth year.

c. To estimate the life expectancy in 1976, we will calculate $t$ as the number of years before 1980 (which will be negative) and substitute it into the linear equation.

To predict the life expectancy of a person born in 2014, we calculate $t$ as the number of years after 1980: $$t = 2014 - 1980 = 34$$ Using the linear model $L = 0.16t + 73.71$: $$L = 0.16 \times 34 + 73.71 = 79.15$$ Thus, the life expectancy of a person born in 2014 is $79.15$ years.

To find the birth year when the life expectancy is 78 years, we set $L = 78$ and solve for $t$: $$78 = 0.16t + 73.71$$ $$0.16t = 78 - 73.71$$ $$t = \frac{78 - 73.71}{0.16} = 26.8125$$ The corresponding birth year is: $$\text{Birth Year} = 1980 + 26.8125 \approx 2007$$ Thus, a person born in 2007 will have a life expectancy of 78 years.

To estimate the life expectancy in 1976, we calculate $t$ as the number of years before 1980: $$t = 1976 - 1980 = -4$$ Using the linear model $L = 0.16t + 73.71$: $$L = 0.16 \times (-4) + 73.71 = 73.07$$ Thus, the life expectancy of a person born in 1976 was $73.07$ years.

Final Answer