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 $(d, C)$ (Spreadsheet at LarsonPrecalculus.com) (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 as ordered pairs $(d, C)$ (Spreadsheet at LarsonPrecalculus.com)
(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

Data Visualization Tools

Data visualization tools, including graphing utilities, are used to plot both the raw data and the mathematical model. This dual plotting allows for the comparison of the observed data points with the predicted values from the inverse variation model, facilitating the assessment of the model’s accuracy and appropriateness in describing the relationship between the variables.

Model Prediction

Model prediction, or extrapolation/interpolation, is the process of using an established mathematical relationship to estimate unknown values. After creating the inverse variation model, one can substitute a specific temperature into the equation to solve for the corresponding depth. This allows for the practical application of the model by providing approximate values for conditions that were not directly measured.

Inverse Variation

Inverse variation describes a relationship where one variable is proportional to the reciprocal of another. In mathematical terms, if y varies inversely with x, then the relationship can be described by y = k/x, where k is a constant. This model implies that as one variable increases, the other decreases at a rate determined by the constant k, assuming the relationship holds true across different data points.

Scatter Plot

A scatter plot is a type of graph used to display values for two variables for a set of data points. In this context, the scatter plot helps visualize the relationship between depth and temperature by plotting each observation as an ordered pair. This visual representation can reveal trends or patterns in the data which might indicate a specific type of relationship, such as a proportional or inverse relationship.

Parameter Estimation

Parameter estimation involves calculating the constant (in this case, k) that best fits the data within a given model. By substituting each pair of data values into the inverse variation model and solving for k, one can obtain several estimates of k. Averaging these estimates is a common technique to determine a representative value that minimizes the effect of variability in the data and ensures the model fits the observed measurements.

Transcript

00:01 Alright, we have these ordered pairs.

00:04 From an oceanographer, you have, it would be your depth and then your temperature in celsius for each one.

00:16 So for each depth, the temperature is recorded in the water.

00:21 Now, you're asked to make a scatterplot of these points, so you should get something similar to this.

00:28 The next thing is, could this variation model accurately represent what's going on in this data? now, if you were to just solve for k, right? so k are constant, would be equal to temperature times of depth.

00:52 If you were to plug in temperature and depth for each one of these five, you would get a different k value, but that doesn't mean that this model won't fit.

01:01 Kind of look at what's happening.

01:02 Here.

01:03 So what this is telling us is that as depth increases, temperature decreases, well, that's clearly what's happening here, right? so it can represent this model.

01:20 So i think yes, that can fit.

01:26 So now what are the k values or the constant values for each order of pair? so if we have them in order from left to right, your first one would be the 1 ,04 .2, and i will actually put these up here under the actual ordered pairs.

01:50 So again, you're taking these numbers and you're plugging them into this equation, and your k values would be 4 ,200, 3 ,800, 4 ,200.

02:11 4 ,800, get rid of that, and 4 ,500.

02:26 So there's, they're not too different.

02:30 So again, that kind of confirms that, yes, the constants are not the same and they should be...

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