An oceanographer took readings of the water temperatures C...

An oceanographer took readings of the water temperatures $C$ (in degrees Celsius) at several depths $d$ (in meters). The data collected are shown in the table. (a) Sketch a scatter plot of the data. (b) Does it appear that the data can be modeled by the inverse variation model $C=k / d ?$ If so, find $k$ for each pair of coordinates. (c) Determine the mean value of $k$ from part (b) to find the inverse variation model $C=k / d$. (d) Use a graphing utility to plot the data points and the inverse model from part (c). (e) Use the model to approximate the depth at which the water temperature is $3^{\circ} \mathrm{C}$.

An oceanographer took readings of the water temperatures $C$ (in degrees Celsius) at several depths $d$ (in meters). The data collected are shown in the table.
(a) Sketch a scatter plot of the data.
(b) Does it appear that the data can be modeled by the inverse variation model $C=k / d ?$ If so, find $k$ for each pair of coordinates.
(c) Determine the mean value of $k$ from part (b) to find the inverse variation model $C=k / d$.
(d) Use a graphing utility to plot the data points and the inverse model from part (c).
(e) Use the model to approximate the depth at which the water temperature is $3^{\circ} \mathrm{C}$.

Key Concepts

Graphing Utility

A graphing utility is a computer or calculator-based tool used to plot data and graphs of mathematical functions. It allows users to visualize the relationships between variables, compare mathematical models to empirical data, and solve equations by providing an interactive, visual representation of the data.

Data Modeling

Data modeling involves creating a mathematical representation of a real-world process or relationship between variables. This approach is used to analyze data, generate predictions, and understand underlying trends or behaviors by fitting a mathematical model, such as an inverse variation model, to observed data.

Mean (Average)

The mean, or average, is a statistical measure that summarizes a set of data by adding all the values and then dividing by the number of values. In the context of modeling, calculating the mean of transformation constants or parameters helps in obtaining a representative value that smooths out individual variances or measurement errors.

Scatter Plot

A scatter plot is a type of graph that displays individual data points on a horizontal and vertical axis to show how two variables are related. It is commonly used to identify trends, patterns, and potential correlations between variables, providing a visual overview of the distribution and variability in the data.

Inverse Variation

Inverse variation describes a relationship between two variables where one variable is proportional to the reciprocal of the other, typically represented by the equation y = k/x. This means that as one variable increases, the other decreases in such a way that their product, denoted by the constant k, remains constant.

Constant of Variation

The constant of variation, symbolized by k in the inverse variation equation, represents the fixed product of the two variables in the relationship. It is a crucial factor in establishing the precise proportional relationship between the variables and determines how strongly they are linked.

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