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Chapter 8: Problem 4

The life expectancies for men and women in the United States can be approximated by the following formulas: $$\begin{aligned} &\text { Women: } E=0.126 t+76.74\\\ &\text { Men: } \quad E=0.169 t+69.11 \end{aligned}$$ where \(E\) represents the length of life in years and \(t\) represents the year of birth, measured as the number of years since 1975 . a. Write a single equation that can be used to determine in what year of birth the life expectancy of men and women would be the same. b. Solve the equation in part a. c. What is the life expectancy for the year of birth determined in part b?

Short Answer

Expert verified
Answer: The life expectancy of men and women in the United States will be the same in approximately the year 2152, and the life expectancy for both will be about 98.95 years.

Step by step solution

01

Set the equations equal to each other

We are trying to find the year of birth where the life expectancies for men and women are the same. Therefore, we equate the two given equations: $$0.126t + 76.74 = 0.169t + 69.11$$
02

Solve for t

Now we need to solve this equation for \(t\). First, isolate the \(t\) variable by moving the terms involving \(t\) to one side and the constant terms to the other side: $$0.169t - 0.126t = 76.74 - 69.11$$ Add the coefficients of \(t\) together and subtract the constants: $$0.043t = 7.63$$ Now, divide both sides of the equation by the coefficient of \(t\) (0.043): $$t = \frac{7.63}{0.043}$$ Calculate the value of \(t\): $$t \approx 177.44$$ Since \(t\) represents the number of years since 1975, we need to add the value of \(t\) to 1975 to find the actual year in which the life expectancies are the same: $$1975 + 177.44 \approx 2152.44$$
03

Find the life expectancy for the year of birth

Since the life expectancies for men and women are the same in this year, we can use either equation to find the length of life (E). We will use the equation for women: $$E = 0.126t + 76.74$$ Insert the value of \(t\) we found in step 2: $$E = 0.126(177.44) + 76.74$$ Now, calculate the value of E: $$E \approx 0.126(177.44) + 76.74 \approx 98.95$$ The life expectancy for the year of birth determined in part b (approximately 2152) is about 98.95 years for both men and women.

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