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Atmospheric pressure: The table below gives a measurement of atmospheric pressure, in grams per square centimeter, at the given altitude, in kilometers.17 $$ \begin{array}{|c|c|} \hline \text { Altitude } & \text { Atmospheric pressure } \\ \hline 5 & 569 \\ \hline 10 & 313 \\ \hline 15 & 172 \\ \hline 20 & 95 \\ \hline 25 & 52 \\ \hline \end{array} $$ (For comparison, 1 kilometer is about 0.6 mile, and 1 gram per square centimeter is about 2 pounds per square foot.) a. Plot the data on atmospheric pressure. b. Make an exponential model for the data on atmospheric pressure. c. What is the atmospheric pressure at an altitude of 30 kilometers? d. Find the atmospheric pressure on Earth’s surface. This is termed standard atmospheric pressure. e. At what altitude is the atmospheric pressure equal to 25% of standard atmospheric pressure?

Step by step solution

01

Plotting the Data

First, plot the given data points on a graph. The x-axis represents altitude in kilometers, and the y-axis represents atmospheric pressure in grams per square centimeter. The given data points are: (5, 569), (10, 313), (15, 172), (20, 95), and (25, 52). Plot each of these points on the graph.

02

Finding an Exponential Model

Assume an exponential model for atmospheric pressure as \( P(h) = P_0 e^{-kh} \), where \( h \) is altitude, \( P_0 \) is the initial pressure (pressure at 0 km), and \( k \) is a decay constant. Using the data, try to fit this model by determining \( P_0 \) and \( k \). Using two points, such as (5, 569) and (25, 52), set up equations and solve them to find \( P_0 \) and \( k \).

03

Calculating the Atmospheric Pressure at 30 km

Using the exponential model \( P(h) = P_0 e^{-kh} \), substitute \( h = 30 \) to find the atmospheric pressure at 30 km. Calculate it using the values found for \( P_0 \) and \( k \).

04

Determining the Atmospheric Pressure at Earth's Surface

To find the atmospheric pressure at sea level (0 km), use the model \( P(0) = P_0 \). With the value of \( P_0 \) previously calculated, this gives us the standard atmospheric pressure.

05

Calculating Altitude for 25% of Standard Atmospheric Pressure

Calculate the altitude where the atmospheric pressure is 25% of the standard pressure (\( 0.25 P_0 \)). Rearrange the exponential formula \( 0.25 P_0 = P_0 e^{-kh} \), to \( 0.25 = e^{-kh} \). Solve for \( h \), using natural logarithms to find \( h = -\frac{\ln(0.25)}{k} \).

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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 on a surface by the weight of the air above that surface in the Earth's atmosphere. It is an essential factor in various scientific fields, including weather forecasting and aerodynamics.

As you go higher in altitude, there is less air above you, and thus, the atmospheric pressure decreases. It is usually measured in terms like grams per square centimeter or pounds per square foot. Atmospheric pressure also affects how comfortable we feel at different altitudes, as it changes the amount of oxygen available. It is vital for students to understand how pressure varies with altitude since it helps in modeling and predicting atmospheric conditions.

Altitude

Altitude refers to how high something is above a reference point, primarily sea level. It plays a crucial role in determining atmospheric pressure. As altitude increases, atmospheric pressure decreases due to the thinning of the atmosphere.

At 5 kilometers, atmospheric pressure might be around 569 grams per square centimeter, but down at sea level, it would be significantly higher. The higher the altitude, the less air exerts pressure upon the surface, which explains why mountain climbers experience breathing difficulties at high altitudes. This relationship between altitude and pressure is non-linear, which is why exponential models are often used to describe it.

Standard Atmospheric Pressure

Standard atmospheric pressure is the atmospheric pressure at sea level and at a temperature of 0 °C (32 °F), which is about 1013.25 millibars or 101325 Pascals.

This value serves as a reference point for assessing other atmospheric pressures. In many scientific and engineering applications, standard atmospheric pressure is used as a baseline to compare the reduction or increase of pressure at various altitudes. Understanding standard atmospheric pressure is vital in calculating how much atmospheric pressure changes with altitude, as seen in the exponential model of atmospheric pressure.

Data Plotting

Data plotting is a method of representing data points on a coordinate system, often using graphs. For the topic at hand, data plotting helps illustrate how atmospheric pressure changes with altitude.

When you plot atmospheric pressure against altitude, the graph typically shows a decreasing curve, indicating exponential decay. This visual aid is crucial for analyzing trends and making predictions.

• The x-axis often represents altitude,
• while the y-axis indicates atmospheric pressure.

By plotting the data points, such as (5, 569) or (25, 52), one can better observe how atmospheric pressure decreases as altitude increases. Data plotting enhances understanding by offering a tangible representation of the relationship between variables.