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 as ordered pairs $(d, C)$ $(1000,4.2) \quad(4000,1.2)$ $(2000,1.9) \quad(5000,0.9)$ $(3000,1.4)$ (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 cach 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 as ordered pairs $(d, C)$
$(1000,4.2) \quad(4000,1.2)$
$(2000,1.9) \quad(5000,0.9)$
$(3000,1.4)$
(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 cach 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

Scatter Plot

A scatter plot is a type of graph used to display values for two variables for a set of data. In this representation, each observation is plotted as a point, which helps reveal any patterns, trends, or correlations between the variables, providing an intuitive visual summary of the data distribution.

Inverse Variation

Inverse variation describes a relationship between two variables in which their product is constant. This means that as one variable increases, the other decreases in such a way that multiplying the two yields the same value, a concept that is foundational in modeling certain types of natural relationships.

Data Modeling

Data modeling involves creating mathematical representations of real-world processes or relationships based on collected data. By choosing an appropriate model, such as an inverse variation model, one can approximate, predict, or understand the behavior of the system under study.

Parameter Estimation

Parameter estimation is the process of using data to determine the constants in a mathematical model. When multiple sets of data are available, finding and averaging the parameter values can improve the model's reliability and accuracy in representing the underlying relationship.

Graphing Utilities

Graphing utilities include software or calculators that allow users to graph functions and data points. They are essential tools for visual analysis, validation of models, and for exploring how well a mathematical model fits the observed data.

Solving for Unknowns Using a Model

Using a mathematical model to solve for an unknown involves substituting a known value into the model equation and solving for the desired variable. This process is vital in prediction and analysis, allowing one to infer unmeasured quantities from established relationships.

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