Social Sciences (Exercises 35–44) Recreational Expenditures...

Social Sciences (Exercises 35–44) Recreational Expenditures Personal consumption expenditures for recreation in billions of dollars in the United States during the years $2000-2013$ can be approximated by the function $$A(t)=632.37 e^{0.0351 t}$$ where $t=0$ corresponds to the year $2000 .$ Based on this model, how much were personal consumption expenditures in 2013 to the nearest billion? (Source: U.S. Bureau of Economic Analysis.)

Social Sciences (Exercises 35–44)
Recreational Expenditures Personal consumption expenditures for recreation in billions of dollars in the United States during the years $2000-2013$ can be approximated by the function
$$A(t)=632.37 e^{0.0351 t}$$
where $t=0$ corresponds to the year $2000 .$ Based on this model, how much were personal consumption expenditures in 2013 to the nearest billion? (Source: U.S. Bureau of Economic Analysis.)

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. These functions are useful for modeling situations where the rate of change of a quantity is proportional to its current value.

Exponential Growth

Exponential growth occurs when a quantity increases by a constant percentage over equal intervals of time. This concept is vital in fields like economics and finance where it describes processes, such as investment returns or population growth, that expand rapidly over time.

Function Evaluation

Function evaluation is the process of substituting a specific value into a function to calculate the corresponding output. This concept is key in applying mathematical models to predict or analyze real-world quantities at particular time points.

Mathematical Modeling

Mathematical modeling involves constructing equations to represent real-world phenomena. In social sciences and economics, models such as exponential functions help in understanding and forecasting trends by capturing the underlying growth dynamics.

Parameter Interpretation

In a mathematical model, parameters represent specific characteristics of the system, such as the initial value and the growth rate. Understanding these parameters is crucial for interpreting the model's behavior and making predictions about future outcomes.

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