Food Shelters (Refer to Example 10 . ) For three food...

Food Shelters $\quad$ (Refer to Example $10 .$ ) For three food shelters, monthly food costs $F$ in dollars, number of people served per month $N$, and monthly charitable receipts $R$ in dollars are as shown in the table. $$ \begin{array}{ccc} \begin{array}{c} \text { Food } \\ \text { Costs }(F) \end{array} & \begin{array}{c} \text { Number } \\ \text { Served }(N) \end{array} & \begin{array}{c} \text { Charitable } \\ \text { Receipts }(R) \end{array} \\ \hline 1300 & 1800 & 5000 \\ 5300 & 3200 & 12,000 \\ 6500 & 4500 & 13,000 \end{array} $$ (a) Model these data using $F=a N+b R+c,$ where $a, b,$ and $c$ are constants. (b) Predict the food costs for a shelter that serves 3500 people and receives charitable receipts of $\$ 12,500$. Round your answer to the nearest hundred dollars.

Food Shelters $\quad$ (Refer to Example $10 .$ ) For three food shelters, monthly food costs $F$ in dollars, number of people served per month $N$, and monthly charitable receipts $R$ in dollars are as shown in the table.
$$
\begin{array}{ccc}
\begin{array}{c}
\text { Food } \\
\text { Costs }(F)
\end{array} & \begin{array}{c}
\text { Number } \\
\text { Served }(N)
\end{array} & \begin{array}{c}
\text { Charitable } \\
\text { Receipts }(R)
\end{array} \\
\hline 1300 & 1800 & 5000 \\
5300 & 3200 & 12,000 \\
6500 & 4500 & 13,000
\end{array}
$$
(a) Model these data using $F=a N+b R+c,$ where $a, b,$ and $c$ are constants.
(b) Predict the food costs for a shelter that serves 3500 people and receives charitable receipts of $\$ 12,500$. Round your answer to the nearest hundred dollars.

Key Concepts

Rounding

Rounding is the process of adjusting a numerical value to a specified degree of precision, typically to simplify the number or to reflect the precision of the data. In applied contexts, such as financial figures or measurements, rounding can make the predicted outcomes more interpretable and practically useful.

Prediction Using Models

Prediction using models involves employing the derived relationship from the data model to estimate the value of the dependent variable for new sets of input values. This process assumes that the model is valid for conditions similar to those observed in the original data, allowing for informed forecasting.

Data Modeling

Data modeling, in this context, refers to the process of creating a mathematical representation of a real-world situation. This includes selecting a functional form (like a linear equation), defining the variables involved, and using observed data to estimate the relationship between these variables, thus enabling analysis and prediction.

System of Linear Equations

A system of linear equations involves equations where each term is either a constant or the product of a constant and a single variable. When multiple data points are available, they can be translated into a system of linear equations which can be solved using methods such as substitution, elimination, or matrix operations to determine the model parameters.

Multiple Linear Regression

Multiple linear regression is a statistical method commonly used to model the relationship between one dependent variable and two or more independent variables by fitting a linear equation to observed data. This approach helps in understanding how each predictor independently influences the outcome, and the combined effect of the predictors explains the variation in the dependent variable.

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