The number of prisoners in state prisons in the United...

The number of prisoners in state prisons in the United States can be approximated by $$ N(x, y)=-260+7 x+2 y-0.009 x y \text { thousand inmates } $$ where $x$ is the number, in thousands, in federal prisons, and $y$ is the number, in thousands, in local jails. ${ }^{6}$ a. In 2007 there were approximately 189 thousand in federal prisons and 781 thousand in local jails. Estimate, to the nearest $0.1$ million, the number of prisoners in state prisons that year. b. Obtain $N$ as a function of $y$ for $x=80$, and again for $x=100$. Interpret the slopes of the resulting linear functions.

The number of prisoners in state prisons in the United States can be approximated by
$$
N(x, y)=-260+7 x+2 y-0.009 x y \text { thousand inmates }
$$
where $x$ is the number, in thousands, in federal prisons, and $y$ is the number, in thousands, in local jails. ${ }^{6}$
a. In 2007 there were approximately 189 thousand in federal prisons and 781 thousand in local jails. Estimate, to the nearest $0.1$ million, the number of prisoners in state prisons that year.
b. Obtain $N$ as a function of $y$ for $x=80$, and again for $x=100$. Interpret the slopes of the resulting linear functions.

Key Concepts

Interpretation of Slope

The slope in the context of a function of one variable represents the rate of change of the output with respect to changes in the input variable. Interpreting the slope involves understanding how each incremental change in the independent variable impacts the dependent variable, which is key in determining marginal effects or sensitivity in applied scenarios.

Reduction to Single Variable Functions

Reducing a multivariable function to a single variable function by fixing one of the variables is a common technique to simplify the analysis. By holding one factor constant, the relationship between the remaining variable and the outcome can be studied in isolation, making it easier to understand how changes in that variable directly influence the result.

Multivariable Functions

A multivariable function is one that has more than one input variable. In such functions, the output depends on combinations of all these inputs. Understanding multivariable functions is essential when analyzing situations where several factors jointly affect the outcome, allowing for a comprehensive interpretation of how each variable contributes to the result.

Function Evaluation with Substitution

Evaluating a function with given data involves substituting specific values for the variables into the function. This technique allows one to determine the output under particular circumstances and is crucial in applying theoretical models to real-world situations.

Unit Conversion

Unit conversion is the process of translating measurements from one scale or set of units to another. This is important when the output of a function is expressed in differing units than those required for analysis or reporting, making it necessary to apply proper conversion factors to ensure the result is in the desired units.

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