We often represent numerical data using tables, diagrams, and graphs. These include various kinds of charts like bar graphs and pie charts, and graphing of functions. We do this to make certain traits of the data easier to notice. Here we will look at how some of these are produced and begin to learn to recognize differences due to rates.
Subsection3.1.1Reading Tables of Data
Sometimes we simply write down the numbers in tables. This works well if we have a limited number of entries and only two aspects to consider. The two aspects become headers for the rows and columns.
Table3.1.1.Stall speed at 2550 lbs, most rearward center of gravity, speeds KIAS
Angle of Bank
Flap Setting
\(0^\circ\)
\(30^\circ\)
\(45^\circ\)
\(60^\circ\)
Up
48
52
57
68
\(10^\circ\)
43
46
51
61
Full
40
43
48
57
Note that stall speed refers to the speed at which a wing will produce insufficient lift to keep a plane flying. It results in the plane lowering its nose to regain speed. Angle of bank refers to how steeply the plane is tipped (left or right) in order to turn. KIAS stands for knots indicated air speed. Indicated airspeed is a speed pilots can see (think speedometer). Flaps are a structure extended for landing and sometimes take-off. Up means they are not in use. Others refer to varying degrees of extension.
Example3.1.2.
What is the stall speed in a \(30^\circ\) bank angle with flaps up?
We can determine this by looking for the column labeled “\(30^\circ\)” and the row labeled “Up”. In that cell is the number 52. Thus the stall speed at that bank angle with flaps up is 52 KIAS.
Example3.1.3.
What is the stall speed in a \(15^\circ\) bank angle with flaps at \(10^\circ\text{?}\)
First, we note that there is no column for \(15^\circ\) bank angle. However we have \(0^\circ\) and \(30^\circ\text{.}\)\(15^\circ\) is half way between these two. For this chart and some others it is reasonable to estimate our desired number by calculating the number between those given.
The two stall speeds are 43 and 46. The number in between (the average) is \((43+46)/2=44.5\text{.}\) When considering stall speeds, it is safest to assume a higher stall speed, so we will round to 45 KIAS.
Checkpoint3.1.4.
What is the stall speed at \(60^\circ\) bank angle and \(10^\circ\) flap setting? KIAS
Answer.
\(61\)
Checkpoint3.1.5.
What is the stall speed at \(37.5^\circ\) bank angle and full flap setting? KIAS
Answer.
\(46\)
Solution.
Note this bank angle is half way between \(30^\circ\) and \(45^\circ\text{.}\) We can average the entries for those bank angles.
Graphs that are curves (like lines) are read by finding a vertical heading that matches our question (think row) and read the corresponding horizontal heading (think column). Note this could be reversed, that is, find a horizontal heading that matches and read the corresponding vertical one.
Example3.1.6.
If the plane with maximum engine out glide represented in Figure 3.1.7 is 2400 ft above the ground how many nautical miles can it glide forward?
Note each horizontal major line is 2000 ft. Counting we find 10 minor lines between each major line, so we know they represent \(2000/10=200\) ft. each vertical major line is 2 nm. Again there are 10 minor lines between each major line, so we know they represent \(2/10=0.2\) nm.
2400 is two minor lines above 2000. We follow that to the blue line, then we follow the gray (minor) line down to the bottom. It is two minor lines before 4. This is \(4-2(0.2)=3.8\) nm.
Figure3.1.7.Graph Representing Maximum Engine Out Glide
Checkpoint3.1.8.
What is the maximum glide for an aircraft 5000 above the ground? nm
Answer.
\(8.2\)
Graphs can be from raw data which can seem random. We still read these the same way.
Example3.1.9.
Figure 3.1.10 has the temperature and dewpoint read by a radiosonde (instruments on weather balloon) as it rose in the atmosphere. Note the vertical axis is the pressure reading. This is not the same as altitude, but it does correspond mostly to altitude. Dew point is the temperature at which water will condense, so it is also a temperature.
What are the temperature and dewpoint at the 700 millibar level?
We follow the 700 mb line over to the dewpoint (green, dashed) line. It is just above \(-20^\circ\) C. We estimate \(-18^\circ\) C. Continuing across the 700 mb line to the temperature (red, solid) line we find it about half way between 0 and 10. We estimate the temperature is \(5^\circ\) C.
Figure3.1.10.Graph of Temperature and Dewpoint
Note some charts like Figure 3.1.10 are not meant to convey specific numbers but rather to show trends.
Example3.1.11.
Clouds form when the temperature reaches the dewpoint. We see in Figure 3.1.10 that the temperature is always higher than the dewpoint, so no clouds are expected where this sounding (weather balloon reading) was taken.
Subsection3.1.3Graphing Discrete Data
Example3.1.12.Increasing Income.
When Vasya was hired in 2017 she was paid an annual salary of $62,347.23. Her work has been good, so each year she has received raises of $5000.00.
To represent this data we first need to calculate her salary each year. We calculate this by iteratively adding the $5000 for each year. This is found in Table 3.1.13
We will represent her salary over time using the bar graph in Figure 3.1.14. Notice the horizontal axis is labeled with years and the vertical axis is labeled in dollars. There is one bar for each year, because her salary was changed only once each year.
Does this graph tell us when Vasya received her raises each year?
If we made the horizontal axes dollars and the vertical axis years, would this change the information presented? Would it be easier or harder to read?
Table3.1.13.Vasya’s Salary
2017
$62,347.23
2018
$67,347.23
2019
$72,347.23
2020
$77,347.23
2021
$82,347.23
Figure3.1.14.Vasya’s Salary
Example3.1.15.A Different Company.
When Tien was hired he was pair an annual salary of $52,429.33. His work has been good so each year he has received raises. His salaries are found in Table 3.1.16. A graph representing his salaries is found in Figure 3.1.17.
How frequently did Tien receive raises?
Can we tell the size of the raises from the graph?
Table3.1.16.Tien’s Salary
2017
$52,429.33
2018
$55,050.80
2019
$57,803.34
2020
$60,693.50
2021
$63,728.18
Figure3.1.17.Tien’s Salary
Subsection3.1.4Graphing Continuous Data
Because the salaries changed only once per year we could use a single entry each year. For other data the changes occur constantly.
Example3.1.18.Scale Model.
A model of a space shuttle is 1:144. This means one inch on the model represents 144 inches on the actual shuttle. Similarly 2 inches on the model represents 288 inches on the actual shuttle. Lengths are not just integers, we could have a part that is 1.72 inches. This would be \(1.72 \cdot 144 = 247.68\) inches on the actual shuttle.
To represent the conversion from inches on the model to inches on the actual object we need to plot points and connect them continuously. A graph of this in Figure 3.1.19. Notice the two points and the extension of those.
\begin{align*}
12 = & I \cdot 8. \\
\frac{12}{8} = & I. \\
1.5 = & I.
\end{align*}
If we reduce the resistance to 4 Ohms we get
\begin{align*}
12 = & I \cdot 4. \\
\frac{12}{4} = & I. \\
3 = & I.
\end{align*}
If instead we increase the resistance to 16 Ohms we get
\begin{align*}
12 = & I \cdot 16. \\
\frac{12}{16} = & I. \\
0.75 = & I.
\end{align*}
We can plot these three points and draw a curve through them. This is found in Figure 3.1.21.
The graph starts with 1 Ohm. Why does it not start at 0?
As the resistance increases what happens to the current?
Figure3.1.21.Graph of Ohms Law
Checkpoint3.1.22.
The ideal gas law expresses a relationship between pressure, volume, and temperature of a gas. It is given by
\begin{equation*}
P \cdot V = k \cdot T
\end{equation*}
where \(P\) is the pressure, \(V\) is the volume, \(T\) is the temperature, and \(k\) is a constant dependent on the specific gas.
(a)
Draw a graph for the equation \(P = \frac{8.3145 T}{2.0000}\text{.}\) Note the units are Kelvin (Celsius + 273.15) for temperature and Jules/litre for pressure. These do not need to be labeled here.
(b)
Draw a graph for the equation \(P = \frac{8.3145 \cdot 293.15}{V}\text{.}\)
Subsection3.1.5Representing Linear Relations
We have seen what linear data looks like in data tables, discrete graphs (e.g., bar graph), and continuous graphs. Here we will add a representation in mathematical symbols that are useful for calculation.
Example3.1.23.Equation for Vasya.
Remember from Example 3.1.12 that Vasya’s annual salary follows a linear pattern. Specifically it increases $5000.00 per year. That means in over one year it has increased $5000.00. Over two years it has increased
Generally then we can say that after \(n\) years, Vasya’s salary has increased \(5000n\) dollars. To know her actual salary we need to add her initial salary. An equation for Vasya’s salary then is
\begin{equation*}
S = \$62,347.23 + \$5000.00 n\text{.}
\end{equation*}
Checkpoint3.1.24.
If Dimitri’s salary is given by the equation
\begin{equation*}
S = \$73,240.00 + \$6000.00 n
\end{equation*}
where \(n\) is the year of employement, what was his initial salary?
What was the size of his annual raises?
Hint.
His initial salary is his salary in year 0.
Answer1.
\(73240\)
Answer2.
\(6000\)
In Example 3.1.18 we graphed a line. In terms of the graph of a line, the rate of change is known as the slope. Because the rate of change is constant, the slope can be calculated from any two points.
Note a rate is a ratio of changes. In terms of points expressed in Cartesian coordinates the calculation is
As expected this is the same slope, because on a line the rate of change (slope) is constant.
Figure3.1.26.Calculating Slope
Checkpoint3.1.27.
Suppose that as dry air rose it dropped in temperature in a linear fashion. If the temperature was measured at 1000 ft MSL as \(21^\circ\) C and at 3000 ft MSL as \(17^\circ\) C, what is the rate of change of temperature with respect to altitude?
What does this imply the temperature is at 0 ft MSL?
Alternately because the temperature drops as the air rises, the temperature at a lower altitude (0 is lower than 1000) will be higher by 2 degrees. So the temperature is \(21+2=23\text{.}\)
