A model that represents the data is given by y=(63.6+0.97 t)/(1+0.01 t), 0 ≤t ≤70 where t is t

step 1 Here the table shown the life expectancy of a child at birth in the US for each of the selected

A model that represents the data is given by y=(63.6+0.97...

A model that represents the data is given by $y=\frac{63.6+0.97 t}{1+0.01 t}, \quad 0 \leq t \leq 70$ where $t$ is the time in years, with $t=0$ corresponding to 1940 (a) Use a graphing utility to graph the data from the table above and the model in the same viewing window. How well does the model fit the data? Explain. (b) Find the $y$ -intercept of the graph of the model. What does it represent in the context of the problem? (c) Use the zoom and trace features of the graphing utility to determine the year when the life expectancy was $70.1 .$ Verify your answer algebraically. (d) Determine the life expectancy in 1978 both graphically and algebraically.

A model that represents the data is given by $y=\frac{63.6+0.97 t}{1+0.01 t}, \quad 0 \leq t \leq 70$
where $t$ is the time in years, with $t=0$ corresponding to 1940
(a) Use a graphing utility to graph the data from the table above and the model in the same viewing window. How well does the model fit the data? Explain.
(b) Find the $y$ -intercept of the graph of the model. What does it represent in the context of the problem?
(c) Use the zoom and trace features of the graphing utility to determine the year when the life expectancy was $70.1 .$ Verify your answer algebraically.
(d) Determine the life expectancy in 1978 both graphically and algebraically.

Key Concepts

Algebraic Problem Solving

Solving equations algebraically is essential for finding specific values of interest within the model. This method includes manipulating the equation to isolate the variable of interest (such as time when life expectancy reaches a particular value) and verifying results, which complements graphical methods and ensures accuracy in predictions.

Intercept Interpretation

The intercept, particularly the y-intercept in this context, represents the beginning value of the dependent variable when the independent variable is zero. In this problem, the y-intercept can be interpreted in the historical context provided (e.g., indicating the life expectancy in the baseline year), offering real-world significance to the mathematical model.

Mathematical Modeling

This concept involves creating and using mathematical formulas to represent real-world phenomena. In this problem, a rational function is used to model the changes in life expectancy over time, which requires an understanding of how model parameters can reflect trends in data and how these models can be validated against observed values.

Rational Functions

Rational functions are ratios of two polynomial functions. They are important in modeling because they can exhibit asymptotic behavior, intercepts, and other features that are useful for approximating nonlinear phenomena. Understanding the structure of these functions helps in interpreting their long-term behavior and determining properties like intercepts.

Graphical Analysis

Graphical analysis involves using tools like graphing utilities to visually compare the mathematical model with actual data points. This approach helps in assessing the model's accuracy, identifying trends, and determining key graph features such as intercepts or specific output values at given input values.

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