Question

Atmospheric pressure The pressure of Earth's atmosphere at sea level is approximately 1000 millibars and decreases exponentially with elevation. At an elevation of \(30,000 \mathrm{ft}\) (approximately the altitude of Mt. Everest), the pressure is one-third the sea-level pressure. At what elevation is the pressure half the sea-level pressure? At what elevation is it \(1 \%\) of the sea-level pressure?

Short answer

Answer: The elevation at which the pressure is half the sea-level pressure is approximately 20,364 ft, and the elevation at which the pressure is 1% of the sea-level pressure is approximately 67,756 ft.

Step-by-step solution

Understanding the given information

First, let's understand the information provided in the problem. We know that the pressure at sea level is 1000 millibars and that it decreases exponentially with elevation. The pressure at an elevation of 30,000 ft is one-third the sea-level pressure.

Define the exponential function

Let's denote the pressure as P (measured in millibars) and the elevation as E (measured in feet). We can write the exponential function as: \(P = P_0 e^{-kE}\) Where \(P_0\) is the initial pressure at sea level, E is the elevation, and k is a constant.

Calculate the constant k

We are given that at an elevation of 30,000 ft, the pressure is one-third the sea-level pressure. Using the exponential function we defined in Step 2, let's plug in the values for P, \(P_0\), and E to find the value of the constant k: \(\frac{P_0}{3} = P_0 e^{-k(30,000)}\) Solving for k, we get: \(k = - \frac{1}{30000} ln(\frac{1}{3})\)

Find the elevation corresponding to half the sea-level pressure

Now, let's use the exponential function with the calculated value of k to find the elevation corresponding to half the sea-level pressure. We want to find E when: \(\frac{P_0}{2} = P_0 e^{-kE}\) Solving for E, we get: \(E = - \frac{1}{k} * ln(\frac{1}{2}) = \frac{30000}{ln(3)} * ln(2) \approx 20,364 \mathrm{ft}\) Thus, the elevation at which the pressure is half the sea-level pressure is approximately 20,364 ft.

Find the elevation corresponding to 1% of the sea-level pressure

In a similar manner, we will find the elevation corresponding to 1% of the sea-level pressure. We want to find E when: \(\frac{P_0}{100} = P_0 e^{-kE}\) Solving for E, we get: \(E = - \frac{1}{k} * ln(\frac{1}{100}) = \frac{30000}{ln(3)} * ln(100) \approx 67,756 \mathrm{ft}\) Thus, the elevation at which the pressure is 1% of the sea-level pressure is approximately 67,756 ft.

Key concepts

Atmospheric Pressure

Atmospheric pressure is the weight of the air above a given point on the Earth's surface. It is the force exerted on an area by the weight of the air columns above it. This pressure plays a significant role in our weather, affecting everything from wind patterns to precipitation.
At sea level, the atmospheric pressure is typically about 1000 millibars. However, as you ascend in elevation, the amount of air above you decreases, which causes the atmospheric pressure to decrease. This reduction in pressure with elevation is due to the thinning of the atmosphere; the higher you go, the less air is above you to exert that pressure.

• Atmospheric pressure is crucial for defining weather and climate conditions.
• It influences physiological processes in humans, such as breathing and blood circulation.
• Various scientific instruments like barometers are used to measure atmospheric pressure.

Sea Level Pressure

Sea level pressure is a standard reference point used for atmospheric pressure measurements. At sea level, the pressure is approximately 1000 millibars. This constant standard is used to compare pressures at different elevations effectively.
To better understand pressure changes with altitude, scientists and meteorologists compare measurements to the pressure at sea level. This way, any changes recorded from higher elevations can be related back to a consistent baseline. Sea level pressure also acts as a crucial reference for calibration in various scientific and industrial settings, such as in aviation.

• Sea level pressure acts as a baseline for pressure comparison globally.
• It helps in creating accurate weather forecasts and mass predictions.
• Different locations, despite having different elevations, can be compared using adjusted sea level pressure readings.

Elevation

Elevation refers to the height above a reference point, such as sea level, and significantly affects atmospheric pressure. The relationship between elevation and pressure is exponential; as elevation increases, pressure decreases at a decreasing rate.
This relationship is mathematically represented by the exponential equation: \[ P = P_0 e^{-kE} \]Here, \( P_0 \) represents the sea level pressure, \( E \) is the elevation, and \( k \) is a constant that depends on the atmospheric conditions.
Understanding how elevation affects pressure is essential for various tasks:

• It is crucial in aviation for understanding aircraft performance with altitude changes.
• Mountain climbers and hikers need to account for lower oxygen levels as they ascend.
• Engineers and designers use these principles in constructing buildings in high-altitude regions.

Changes in atmospheric pressure with elevation underscore the importance of factoring in elevation in scientific studies, weather prediction, and various industrial applications. This relationship helps in planning and survival elements connected with high-altitude environments.