Solve each problem. See Example 11. Atmospheric Pressure The...

Solve each problem. See Example 11. Atmospheric Pressure The atmospheric pressure (in millibars) at a given altitude (in meters) is shown in the table. $$\begin{array}{c|c||c|c}\text { Altitude } & \text { Pressure } & \text { Altitude } & \text { Pressure } \\ \hline 0 & 1013 & 6000 & 472 \\\hline 1000 & 899 & 7000 & 411 \\\hline 2000 & 795 & 8000 & 357 \\ \hline 3000 & 701 & 9000 & 308 \\\hline 4000 & 617 & 10,000 & 265 \\\hline 5000 & 541 & & \\ \hline\end{array}$$ (a) Use a graphing calculator to make a scatter diagram of the data for atmospheric pressure $P$ at altitude $x$ (b) Would a linear or an exponential function fit the data better? (c) The following function approximates the data. $$P(x)=1013 e^{-0.0001341 x}$$ Use a graphing calculator to graph $P$ and the data on the same coordinate axes. (d) Use $P$ to predict the pressures at $1500 \mathrm{m}$ and $11,000 \mathrm{m},$ and compare them to the actual values of 846 millibars and 227 millibars, respectively.

Solve each problem. See Example 11.
Atmospheric Pressure The atmospheric pressure (in millibars) at a given altitude (in meters) is shown in the table.
$$\begin{array}{c|c||c|c}\text { Altitude } & \text { Pressure } & \text { Altitude } & \text { Pressure } \\
\hline 0 & 1013 & 6000 & 472 \\\hline 1000 & 899 & 7000 & 411 \\\hline 2000 & 795 & 8000 & 357 \\
\hline 3000 & 701 & 9000 & 308 \\\hline 4000 & 617 & 10,000 & 265 \\\hline 5000 & 541 & & \\
\hline\end{array}$$
(a) Use a graphing calculator to make a scatter diagram of the data for atmospheric pressure $P$ at altitude $x$
(b) Would a linear or an exponential function fit the data better?
(c) The following function approximates the data.
$$P(x)=1013 e^{-0.0001341 x}$$
Use a graphing calculator to graph $P$ and the data on the same coordinate axes.
(d) Use $P$ to predict the pressures at $1500 \mathrm{m}$ and $11,000 \mathrm{m},$ and compare them to the actual values of 846 millibars and 227 millibars, respectively.

Key Concepts

Prediction and Extrapolation

Prediction involves using a mathematical model to estimate values of a variable at points where data has not been explicitly provided. Extrapolation is the process of extending the model beyond the observed data range to forecast future or unmeasured values, which is critical in applications where direct measurement may be challenging or impossible.

Graphing Calculators for Visualization

Graphing calculators are tools that allow for the plotting of data points and functions on the same coordinate system. They are especially useful for visually comparing mathematical models with actual data, verifying the accuracy of the function's fit, and facilitating the analysis of trends and predictions.

Model Comparison and Selection

Model comparison involves evaluating different types of functions, such as linear versus exponential, to determine which one better represents the behavior of the data. This process includes comparing how well each model fits the scatter plot and assessing error measures, guiding the choice of the most suitable model for predictions.

Scatter Diagram

A scatter diagram is a graphical representation of paired data points where one variable is plotted along the horizontal axis and the other along the vertical axis. It helps in visualizing the pattern or relationship between the two variables, making it easier to judge whether a particular type of function might fit the data well.

Exponential Function Modeling

Exponential functions are used to model processes that change at rates proportional to their current value, such as decay or growth phenomena. In contexts like atmospheric pressure change with altitude, where the rate of decrease is proportional to the existing pressure, exponential functions are often appropriate and can provide accurate approximations for the observed data.

Transcript

00:01 So for this problem, we are given a chart of altitudes and pressures in meters and in millivars respectively.

00:08 And using this data, we have to complete the questions below.

00:11 So first, we have to make a scatterplot of the data we are given.

00:15 So for this, you'll need a grader later.

00:18 I like desmos .com because it makes it really easy to put tables in, but you do whatever option works best for you.

00:25 So i'm going to start by putting in the x values.

00:28 So we start with 0, 1 ,000, 2 ,000, 3 ,000, 4 ,000, 5 ,000, 6 ,000, all the way to 10 ,000.

00:39 Now for the y values, we start at 1013, 899, then we go to 795, 701, 617, 541, 472, 411, 357, 308, and then and last, 265.

01:11 So this is essentially part a.

01:12 We've made our scatter plot onto the graph, and you can see it makes this nice little crest.

01:18 So in 2, we have to determine if a linear exponential function would be better for the graph we just made.

01:25 So you'll see that it's not a straight line.

01:28 Even if you look at the numbers, you can see that the difference between the first two wire values, 1013 and 899, is roughly about 110.

01:38 But if you look at the last two values, it's nowhere near that.

01:42 You can even see by looking at the graph itself, the difference between these two points right here in the y values.

01:49 So it is for this reason, an exponential function would fit this graph better.

01:56 So i'm going to write for b, it will be exponential.

02:07 Now for part c, we'll be returning to our graph and we are going to be plotting this equation.

02:14 So once again, all it is is just 1013e to the power of negative 0 .000 -13 -1 -13 -x.

02:28 And you can see it makes a nice little line, and even more so, you can see it almost matches the scatter plot that we made.

02:36 So this would be what we do for part c...

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