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Chapter 13: Problem 36

$$\text {Plot the indicated graphs.}$$ The atmospheric pressure \(p\) (in kPa) at a given altitude \(h\) (in \(\mathrm{km}\) ) is given in the following table. On semilog paper, plot \(p\) as a function of \(h\) $$\begin{array}{l|c|c|c|c|c} h(\mathrm{km}) & 0 & 10 & 20 & 30 & 40 \\ \hline p(\mathrm{kPa}) & 101 & 25 & 6.3 & 2.0 & 0.53 \end{array}$$

Short Answer

Expert verified
The graph of pressure vs. altitude on semilog paper will show an exponential decay curve, decreasing as altitude increases.

Step by step solution

01

Understand the Problem

We need to plot the atmospheric pressure, \(p\), as a function of altitude, \(h\), using semilogarithmic paper. This requires plotting the given table values, with \(h\) on the regular linear scale and \(p\) on the logarithmic scale.
02

Examine the Data

Review the table given with \(h\) (in km) and \(p\) (in kPa): \(h = [0, 10, 20, 30, 40]\) and \(p = [101, 25, 6.3, 2.0, 0.53]\). We observe that as altitude increases, pressure decreases, confirming it's an inverse relationship.
03

Prepare the Semilog Graph Paper

On semilog graph paper, the vertical axis (y-axis) should use a logarithmic scale for \(p\). The horizontal axis (x-axis) will remain a linear scale representing \(h\). Choose appropriate scales to fit all data points on your graph.
04

Plot the Data Points

Plot each pair \((h, p)\) on the semilog paper. For example, at \(h = 0\) km, plot \(p = 101\) kPa; at \(h = 10\) km, plot \(p = 25\) kPa, and so on until all points are plotted. Make sure to verify that each \(p\) value corresponds correctly on the logarithmic scale.
05

Connect the Points

After plotting all the points, connect them with a smooth curve. The plot should reflect the exponential decline in pressure as altitude increases.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Atmospheric Pressure
Atmospheric pressure is the force exerted by the weight of the air above a surface. It is measured in units such as kilopascals (kPa), and its value decreases as you go higher up from the sea level. Air is dense at lower altitudes, which results in higher atmospheric pressure. At sea level, the standard atmospheric pressure is about 101 kPa or 1 atmospheric pressure unit (atm).

Understanding atmospheric pressure is crucial because it affects various weather patterns and phenomena. For instance:
  • High-pressure areas usually bring clear skies.
  • Low-pressure areas are associated with clouds and precipitation.
Knowing about atmospheric pressure can also help predict weather changes, making it a vital factor in meteorology.
Altitude and Pressure Relationship
The relationship between altitude and atmospheric pressure is inversely proportional. As the altitude increases, the atmospheric pressure decreases. This occurs because there is less air above the surface when you are at higher altitudes.

At higher altitudes:
  • The amount of air and therefore its weight decreases.
  • As the air becomes less dense, the atmospheric pressure drops.
For example, at sea level (0 km), the pressure is around 101 kPa, but at 10 km, it falls drastically to about 25 kPa.
This inverse relationship can be represented mathematically by an exponential function, which is typically used to model how atmospheric pressure decreases with altitude.
Plotting Data on Semilog Paper
Plotting data on semilogarithmic (semilog) paper is a helpful way to visualize relationships like that of pressure and altitude. In a semilog plot, one axis is logarithmic and the other is linear. For the problem at hand, atmospheric pressure (\(p\) in kPa) is plotted on the logarithmic axis, while altitude (\(h\) in km) is on the linear axis.

The advantages of using semilog paper include:
  • It allows you to see exponential relationships more clearly.
  • It handles large variations in values with ease, making it suitable for data that span several orders of magnitude, like atmospheric pressure at varying altitudes.
To plot the data:
  • Locate each altitude on the x-axis (linear scale).
  • Find the corresponding pressure on the y-axis (logarithmic scale) and plot the point.
After plotting all points, you draw a curve connecting them, which will typically show a neat exponential decay, reflecting how pressure decreases with increasing altitude.

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Most popular questions from this chapter

Perform the indicated operations. In Exercise \(47,\) the graph of \(y=\log _{0.5} x\) is plotted. By inspecting the graph and noting the properties of \(\log _{0.5} x,\) describe some of the differences of logarithms to a base less than 1 from those to a base greater than 1.

\(\text {Solve the given problems.}\) When a person ingests a medication capsule, it is found that the rate \(R\) (in \(\mathrm{mg} / \mathrm{min}\) ) that it enters the bloodstream in time \(t\) (in \(\min\) ) is given by \(\log _{10} R-\log _{10} 5=t \log _{10} 0.95 .\) Solve for \(R\) as a function of \(t\)

$$\text {Plot the indicated graphs.}$$ One end of a very hot steel bar is sprayed with a stream of cool water. The rate of cooling \(R\) (in \(^{\circ} \mathrm{F} / \mathrm{s}\) ) as a function of the distance \(d\) (in in.) from one end of the bar is then measured, with the results shown in the following table. On log-log paper, plot \(R\) as a function of \(d .\) Such experiments are made to determine the hardness of steel. $$\begin{aligned} &\begin{array}{l|l|l|l|l} d \text { (in.) } & 0.063 & 0.13 & 0.19 & 0.25 \\ \hline R\left(^{\circ} \mathrm{F} / \mathrm{s}\right) & 600 & 190 & 100 & 72 \end{array}\\\ &\begin{array}{l|c|c|c|c|c} d \text { (in.) } & 0.38 & 0.50 & 0.75 & 1.0 & 1.5 \\ \hline R\left(^{\circ} \mathrm{F} / \mathrm{s}\right) & 46 & 29 & 17 & 10 & 6.0 \end{array} \end{aligned}$$

Use a calculator to solve the given equations. According to one model, the number \(N\) of Americans (in millions) age 65 and older that will have Alzheimer's disease \(t\) years after 2015 is given by \(N=5.1(1.03)^{t} .\) In what year will this number reach 8.0 million?

Use a calculator to solve the given equations. \(15^{-x}=1.326\)
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