Problem 57 - Calculus with Concepts in Calculus

Question

Halley's Law states that the barometric pressure \(p(t)\) in inches of mercury at \(t\) miles above sea level is given by $$ p(t) \approx 29.92 e^{-0.2 t} \quad \text { for } t \geq 0 $$ Find the barometric pressure a. at sea level b. 5 miles above sea level c. 10 miles above sea level

Step-by-Step Solution

Answer

a. 29.92 in; b. 11.00 in; c. 4.05 in.

Step 1: Understand the Formula

Halley's Law is given by the formula for barometric pressure: \( p(t) \approx 29.92 e^{-0.2 t} \), where \( t \) is the height in miles above sea level. The formula decreases exponentially as \( t \) increases.

Step 2: Calculate Pressure at Sea Level

To find the barometric pressure at sea level, we substitute \( t = 0 \) into the equation: \( p(0) \approx 29.92 e^{-0.2 \times 0} = 29.92 e^{0} = 29.92 \). So, the pressure at sea level is 29.92 inches of mercury.

Step 3: Calculate Pressure 5 Miles Above Sea Level

Substitute \( t = 5 \) into the equation: \( p(5) \approx 29.92 e^{-0.2 \times 5} = 29.92 e^{-1} \). We know that \( e^{-1} \approx 0.3679 \), so \( p(5) \approx 29.92 \times 0.3679 \approx 11.00 \). Thus, the pressure 5 miles above sea level is approximately 11.00 inches of mercury.

Step 4: Calculate Pressure 10 Miles Above Sea Level

Substitute \( t = 10 \) into the equation: \( p(10) \approx 29.92 e^{-0.2 \times 10} = 29.92 e^{-2} \). Knowing that \( e^{-2} \approx 0.1353 \), we find \( p(10) \approx 29.92 \times 0.1353 \approx 4.05 \). Therefore, the pressure 10 miles above sea level is approximately 4.05 inches of mercury.

Key Concepts

Barometric Pressure

Barometric pressure is a measure of the weight of air above a specific point on Earth's surface. It is often expressed in inches of mercury, representing the height of a column of mercury balanced under the atmospheric pressure. This measurement is crucial for meteorology and aviation predictions.

The formula used in Halley's Law, \(p(t) = 29.92 e^{-0.2t}\), captures the change in barometric pressure with altitude. At sea level, where \(t = 0\), the natural exponential function \(e^0\) equals 1, making the pressure equal to 29.92 inches of mercury. As you ascend, the air becomes less dense and exerts less pressure.

This exponential decay demonstrates how rapidly barometric pressure drops with increased altitude. It is a valuable insight for understanding why high altitudes might require adjustments for breathing and flying aircraft safely.

Halley's Law

Halley's Law provides a simplified model for understanding changes in barometric pressure with altitude. Named after the astronomer Edmond Halley, the law approximates the pressure experienced at various heights above sea level using an exponential function.

In the case of Halley's equation, the coefficient 29.92 represents the standard atmospheric pressure at sea level. The factor \(e^{-0.2t}\) illustrates the rate at which pressure decreases as altitude rises. This aspect of exponential decay is significant because it indicates that pressure decreases at a diminishing rate.

For example, calculated pressures at 5 and 10 miles above sea level using Halley's Law demonstrate substantial decreases in atmospheric pressure. At 5 miles, the pressure is about 11.00 inches of mercury, while at 10 miles, it drops even further to approximately 4.05. These insights help to predict environmental conditions in various layers of the atmosphere.

Calculus Applications

Calculus provides essential tools for analyzing and predicting how functions like Halley's Law behave over a range of variables. In this scenario, calculus reveals the exponential decay of barometric pressure with altitude. The derivative can further disclose the rate of change in pressure at any given altitude.

Understanding the behavior of \(p(t) = 29.92 e^{-0.2t}\) is key, as it tells us how pressure decreases not just numerically, but also conceptually. By calculating the derivative \(p'(t)\), one could identify how rapidly pressure changes with a small increase in height. This application has profound implications for fields like aerospace, where precise measurement and prediction of atmospheric conditions are critical.

In summary, calculus enhances our understanding of exponential decay by not only quantifying this natural phenomenon, but also providing insight into how it progresses, aiding in the creation of accurate models like Halley’s Law for practical applications.