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Chapter 4: Problem 31

Atmospheric pressure Under certain conditions the atmospheric pressure \(p\) (in inches) at altitude \(h\) feet is given by \(p=29 e^{-0.000034 h}\). What is the pressure at an altitude of A. \(30,000\) feet? B. \(40,000\) feet?

Short Answer

Expert verified
At 30,000 feet, pressure is approx. 10.48 inches; at 40,000 feet, it's approx. 7.45 inches.

Step by step solution

01

Understand the Formula

The given formula for atmospheric pressure is \( p = 29 e^{-0.000034 h} \), where \( p \) is the pressure in inches and \( h \) is the altitude in feet. Our task is to substitute different values for \( h \) to find pressure at specified altitudes.
02

Calculation for 30,000 feet

Substitute \( h = 30,000 \) feet into the formula: \( p = 29 e^{-0.000034 \, \times \, 30000} \). First calculate the exponent: \(-0.000034 \, \times \, 30000 = -1.02\). Next, compute \( e^{-1.02} \). Finally, multiply the result by 29 to find \( p \).
03

Compute for 30,000 feet with e-evaluation

Evaluating \( e^{-1.02} \) gives approximately 0.3615. So, \( p = 29 \, \times \, 0.3615 \approx 10.4835 \). The atmospheric pressure at 30,000 feet is approximately 10.48 inches.
04

Calculation for 40,000 feet

Substitute \( h = 40,000 \) feet into the formula: \( p = 29 e^{-0.000034 \, \times \, 40000} \). Calculate the exponent: \(-0.000034 \, \times \, 40000 = -1.36\). Then, compute \( e^{-1.36} \). Multiply the result by 29 to find \( p \).
05

Compute for 40,000 feet with e-evaluation

Evaluating \( e^{-1.36} \) gives approximately 0.2570. So, \( p = 29 \, \times \, 0.2570 \approx 7.453 \). The atmospheric pressure at 40,000 feet is approximately 7.45 inches.

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Key Concepts

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

Exponential Functions
Exponential functions are a fundamental concept in mathematics, often used to describe situations where quantities grow or decay at a rate proportional to their current value. The general form of an exponential function is \( y = ab^{x} \), where:
  • \( a \) is the initial amount or y-intercept
  • \( b \) is the base or factor which determines the direction and rate of change
  • \( x \) is the exponent
In our exercise, we deal with the exponential function \( p = 29 e^{-0.000034 h} \), where:
  • \( 29 \) is the initial pressure value
  • \( e \) is the base, Euler's number, approximately 2.718
  • \( -0.000034 h \) is the exponent, which changes based on altitude \( h \)
Understanding this function is crucial as it signifies how the pressure decreases exponentially with increasing altitude. As \( h \) increases, the exponent becomes more negative, thus reducing the pressure \( p \) rapidly.
Altitude and Pressure Relationship
Atmospheric pressure and altitude have an inverse relationship, which means as one increases, the other decreases. The mathematical representation \( p = 29 e^{-0.000034 h} \) shows this exponential decay of atmospheric pressure with increasing height. To visualize this:
  • At sea level (\( h = 0 \)), pressure is at its maximum with \( p = 29 \) inches.
  • At higher altitudes like 30,000 feet and 40,000 feet, \( p \) has decreased significantly, reflecting how air becomes thinner the higher one goes.
This relationship is vital in fields like aviation and meteorology, helping predict weather patterns and design aircraft capable of operating at various altitudes. Pilots and scientists use such formulas to ensure safe and efficient flights.
Evaluating Expressions
Evaluating expressions involves substituting values into a mathematical formula to determine a specific outcome. This is done systematically and requires a thorough understanding of mathematical operations. Let's explore the key steps:
  • Substitute the given value into the expression. For example, to find pressure at 30,000 feet, substitute \( h = 30,000 \) into \( p = 29 e^{-0.000034 h} \).
  • Calculate the exponent. Here, compute \(-0.000034 \times 30000 = -1.02\).
  • Evaluate the exponential term using \( e \). For \( e^{-1.02} \), use a calculator or mathematical software to find approximately 0.3615.
  • Finally, multiply by the initial constant to obtain the result, \( p = 29 \times 0.3615 \approx 10.48 \).
The same steps apply for any other altitude like 40,000 feet. Through evaluating such expressions, one can quickly determine the impacts of changes in altitude on atmospheric conditions, a necessary competency in various scientific and practical applications.

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Most popular questions from this chapter

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