Solve each problem. Atmospheric Pressure The atmospheric...

Solve each problem. Atmospheric Pressure The atmospheric pressure (in millibars) at a given altitude (in meters) is shown in the table. $$\begin{array}{c|c|c|c} \text { Altitude } & \text { Pressure } & \text { Altitude } & \text { Pressure } \\ \hline 0 & 1013 & 6000 & 472 \\ 1000 & 899 & 7000 & 411 \\ 2000 & 795 & 8000 & 357 \\ 3000 & 701 & 9000 & 308 \\ 4000 & 617 & 10,000 & 265 \\ 5000 & 541 & & \end{array}$$ (a) Use a graphing calculator to make a scatter diagram of the data for atmospheric pressure $P$ at altitude $x$ (b) Use the concept of average rate of change to determine whether a linear or exponential function would fit the data better. (c) The function $$ P(x)=1013 e^{-0.0001341 x} $$ approximates the data. 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 with the actual values of 846 millibars and 227 millibars, respectively.

Solve each problem.
Atmospheric Pressure The atmospheric pressure (in millibars) at a given altitude (in meters) is shown in the table.
$$\begin{array}{c|c|c|c}
\text { Altitude } & \text { Pressure } & \text { Altitude } & \text { Pressure } \\
\hline 0 & 1013 & 6000 & 472 \\
1000 & 899 & 7000 & 411 \\
2000 & 795 & 8000 & 357 \\
3000 & 701 & 9000 & 308 \\
4000 & 617 & 10,000 & 265 \\
5000 & 541 & &
\end{array}$$
(a) Use a graphing calculator to make a scatter diagram of the data for atmospheric pressure $P$ at altitude $x$
(b) Use the concept of average rate of change to determine whether a linear or exponential function would fit the data better.
(c) The function
$$
P(x)=1013 e^{-0.0001341 x}
$$
approximates the data. 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 with the actual values of 846 millibars and 227 millibars, respectively.

Key Concepts

Scatter Diagram

A scatter diagram is a type of graph used to display individual data points on a coordinate system, with one variable along the horizontal axis and another along the vertical axis. This graphical tool helps in visualizing the potential relationship or correlation between two variables, identifying trends and patterns, and assessing the spread of the data.

Average Rate of Change

The average rate of change measures how a quantity changes, on average, over a given interval of the independent variable. It is calculated by taking the difference in the dependent variable divided by the difference in the independent variable between two points, providing insight into the overall trend of the data and helping to decide on an appropriate model for the relationship.

Linear and Exponential Models

Linear and exponential models are two common types of mathematical models used to describe relationships between variables. A linear model implies a constant rate of change, represented by a straight line, while an exponential model implies a multiplicative rate of change, where the quantity increases or decreases proportionally to its current value, often visualized as a curved graph.

Exponential Decay

Exponential decay is a process where a quantity decreases at a rate proportional to its current value, often modeled by equations of the form y = y?e^(-kx). This concept is widely used in various fields to describe phenomena that diminish over time or distance, such as the decrease in atmospheric pressure with increasing altitude.

Predictive Modeling

Predictive modeling involves creating a mathematical model based on data that can be used to estimate or forecast unknown values. The model is fitted to the available data and then applied to make predictions at new input values, which can later be compared with real observations or measurements to assess the model's accuracy.

Transcript

00:01 So hello student now we are going to understand this question.

00:02 This question is about atmospheric pressure that massive pressure in mini bar at the given altitude is shown in the table.

00:13 The first part a huge graphing calculated to make the scattered diagram of the data for the atmospheric pressure at altitude so let us make graphic calculator graph so this is a scattered diagram of the given data of atmosphere p at altitude this is pressure and this is altitude.

01:08 This is pressure on the y -axis and this is altitude.

01:29 Now part b.

01:32 Use the concept of average rate of changing the determine whether the linear or exponent function would be fit for data.

01:42 So it is exponential decreasing function.

01:46 So and exponential function fit better because the average rate the average rate of change between data and point is not constant constant.

02:41 Now see the function is given px equal to 1 .13 and e to the power minus 0 .0 .03 0 .03 .0 .03.

03:13 X.

03:16 Examate the data use of graphing calculated to graph p the data on the same coordinate axis.

03:23 The graph is, if you draw the graph, graph, let us see on this only.

03:33 The graph is similar.

03:36 Same, basically this is the graph of the same.

03:39 So this is the graph for point c and the d part.

04:01 Use p to predict the pressure rate 15 mm, 11 ,000 m compared with the actual value...

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