Question

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 $(d, C)$. $$\begin{aligned} &(1000,4.2) \quad(4000,1.2)\\ &(2000,1.9) \quad(5000,0.9)\\ &(3000,1.4) \end{aligned}$$ (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 $(d, C)$.
$$\begin{aligned}
&(1000,4.2) \quad(4000,1.2)\\
&(2000,1.9) \quad(5000,0.9)\\
&(3000,1.4)
\end{aligned}$$
(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 $3^{\circ} \mathrm{C}$.

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Added by Sofia C.

Precalculus with Limits

Ron Larson 2nd Edition

Instant Answer

Solved by Expert Eric Carlsen

07/01/2022

Step 1

2), (2000, 1.9), (3000, 1.4), (4000, 1.2), and (5000, 0.9) on a graph. Show more…

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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 $(d, C)$. $$\begin{aligned} &(1000,4.2) \quad(4000,1.2)\\ &(2000,1.9) \quad(5000,0.9)\\ &(3000,1.4) \end{aligned}$$ (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 $3^{\circ} \mathrm{C}$.

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Transcript

00:03 In this problem, we are given data that describes or relates the depth and meters to the temperature of the water in degrees celsius.

00:13 And we are asked to plot this information on the coordinate plane.

00:18 So we have gone right 1 ,000 and up 4 .2, then write 2 ,000, up 1 .9, write 3 ,000 up 1 .4, write 4 ,000 up 1 .2, and write 5 ,000 up 0 .9.

00:34 So here are our points.

00:35 And we're asked, does the data match the model c equals k over d? well, this is inverse variation or indirect variation.

00:54 And indirect variation has a model that looks like this as one of our variables increases, the other decreases.

01:01 So that is definitely happening here.

01:03 And we are then asked to find k by doing an average of all of the k values here.

01:10 So what we know here in indirect variation, y equals k to the x or c equals k over d.

01:19 So if we solve this, we get k is equal to the x times the y.

01:24 So if we multiply our x and our y values together.

01:27 So 1 ,000 times 4 .2 and 2 ,000 times 1 .9.

01:35 And 3 ,000 times 1 .2, 4 ,000.

01:40 Sorry, 3 ,000 times 1 .4 ,000 times 1 .2, and 5 ,000 times 0 .9, we get these values right here.

01:53 So if we take all of these up and add them together, the sum here comes out to be 21 ,500...

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