Beef consumption in the United States (in billions of...

Beef consumption in the United States (in billions of pounds) in year $x$ can be approximated by the function $$ f(x)=-154.41+39.38 \ln x \quad(x \geq 90) $$ where $x=90$ corresponds to $1990 .$ (a) How much beef was consumed in 1999 and in $2002 ?$ (b) According to this model when will beef consumption reach 35 billion pounds per year?

Beef consumption in the United States (in billions of pounds) in year $x$ can be approximated by the function
$$
f(x)=-154.41+39.38 \ln x \quad(x \geq 90)
$$
where $x=90$ corresponds to $1990 .$
(a) How much beef was consumed in 1999 and in $2002 ?$
(b) According to this model when will beef consumption reach 35 billion pounds per year?

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Key Concepts

Logarithmic Functions

Logarithmic functions, particularly the natural logarithm, are used to model situations where growth or change occurs at a decreasing rate. Their characteristic slow increase makes them suitable for modeling phenomena that do not change linearly over time.

Function Evaluation

Function evaluation involves substituting specific input values into a function to determine the corresponding outputs. This basic skill is essential in applying any mathematical model to real-world data by determining the model's prediction at particular points.

Solving Equations Involving Logarithms

When a logarithmic function is set equal to a particular value, solving the equation typically requires isolating the logarithmic term and then applying the properties of logarithms and exponentials. This process is critical for finding the input value that yields a specific output in the model.

Mathematical Modeling and Extrapolation

Mathematical modeling uses functions to approximate real-world phenomena. Extrapolation involves using the model to predict values outside the range of the original data. Understanding the assumptions and limitations of the model is important when making such predictions.

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