The per capita consumption (in pounds) of all beef in the...

The per capita consumption (in pounds) of all beef in the United States is given by the function $C(x)=-0.33 x+67.1,$ where $x$ is the number of years since $2000 .$ (Source: Based on data from the Economic Research Service, U.S. Department of Agriculture, $2000-2008$ ) a. Find and interpret $C(4)$. b. Estimate the per capita consumption of beef in the United States in $2010 .$

The per capita consumption (in pounds) of all beef in the United States is given by the function $C(x)=-0.33 x+67.1,$ where $x$ is the number of years since $2000 .$ (Source: Based on data from the Economic Research Service, U.S. Department of Agriculture, $2000-2008$ )
a. Find and interpret $C(4)$.
b. Estimate the per capita consumption of beef in the United States in $2010 .$

Key Concepts

Linear Function

A linear function is an algebraic expression that describes a straight-line relationship between two variables. It is typically written in the form y = mx + b, where m represents the constant rate of change (slope) and b represents the initial value when the independent variable is zero (y-intercept). This concept is central in modeling relationships with a constant rate of change.

Function Evaluation

Function evaluation is the process of substituting a specific value into a function to determine the corresponding output. This is fundamental in understanding how changes in the input affect the output, particularly in applied mathematics where functions model real-world situations.

Interpreting Linear Models

Interpreting linear models involves analyzing the components of the linear function (slope and intercept) to understand the context of the data. By interpreting the slope, one can determine the rate at which the dependent variable changes with respect to the independent variable, and by looking at the intercept, one understands the starting value of the dependent variable when the independent variable is zero.

Extrapolation

Extrapolation is the method of extending the known data beyond its current range to make predictions about future or past values. It involves using the established trend of a function, such as a linear function, to estimate values at points not originally observed, which is particularly useful in forecasting scenarios.

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