Problem 45 - Intermediate Algebra

Question

The atmospheric pressure, measured in pounds per square inch, is a function of the altitude above sea level. The equation \(P(a)=14.7 e^{-0.21 a}\), where \(a\) is the altitude measured in miles, can be used to approximate atmospheric pressure. Find the atmospheric pressure at each of the following locations. Express each answer to the nearest tenth of a pound per square inch. (a) Mount McKinley in Alaska-altitude of \(3.85\) miles (b) Denver, Colorado-the "mile-high" city (c) Asheville, North Carolina-altitude of 1985 feet (5280 feet \(=1\) mile \()\) (d) Phoenix, Arizona-altitude of 1090 feet

Step-by-Step Solution

Answer

(a) 6.8 psi, (b) 12.1 psi, (c) 13.6 psi, (d) 14.1 psi.

Step 1: Apply Formula to Mount McKinley

To find the atmospheric pressure at Mount McKinley, where the altitude is 3.85 miles, use the formula: \( P(a) = 14.7 \times e^{-0.21 \times 3.85} \). Calculate \( -0.21 \times 3.85 = -0.8085 \). Thus, \( P(3.85) = 14.7 \times e^{-0.8085} \). Evaluating this gives approximately 6.8 pounds per square inch.

Step 2: Apply Formula to Denver

Denver is known as 'mile-high' due to its altitude of 1 mile above sea level. Use the formula: \( P(a) = 14.7 \times e^{-0.21 \times 1} \). Calculate \( -0.21 \times 1 = -0.21 \). Thus, \( P(1) = 14.7 \times e^{-0.21} \). Evaluating this gives approximately 12.1 pounds per square inch.

Step 3: Convert Feet to Miles for Asheville

Asheville's altitude is given in feet, so first convert 1985 feet to miles: \( 1985 \div 5280 \approx 0.3765 \) miles. Use this in the formula: \( P(a) = 14.7 \times e^{-0.21 \times 0.3765} \). Calculate \( -0.21 \times 0.3765 \approx -0.0791 \). Thus, \( P(0.3765) = 14.7 \times e^{-0.0791} \). Evaluating this gives approximately 13.6 pounds per square inch.

Step 4: Convert Feet to Miles for Phoenix

Phoenix's altitude is given in feet, so convert 1090 feet to miles: \( 1090 \div 5280 \approx 0.2068 \) miles. Use this in the formula: \( P(a) = 14.7 \times e^{-0.21 \times 0.2068} \). Calculate \( -0.21 \times 0.2068 \approx -0.0434 \). Thus, \( P(0.2068) = 14.7 \times e^{-0.0434} \). Evaluating this gives approximately 14.1 pounds per square inch.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions involving a constant base raised to a variable exponent. In our exercise, the function \( P(a) = 14.7 \times e^{-0.21a} \) is an exponential equation used to model atmospheric pressure in relation to altitude. Here, \( e \) is the mathematical constant approximately equal to 2.71828, which serves as the base of the natural logarithm. When working with exponential functions, the exponent plays a crucial role, determining how rapidly the function grows or decays.

• The expression \( -0.21a \) in the exponent indicates an exponential decay as altitude \( a \) increases. This means that pressure decreases quickly as you go higher into the atmosphere.
• The negative sign signifies that as the altitude \( a \) grows, the exponent becomes more negative, and \( e^{-0.21a} \) produces a smaller value, thereby reducing the atmospheric pressure.

Understanding these properties helps us compute pressure changes with altitude efficiently.

Altitude Conversion

Altitude conversion is vital in problems where height measurements are given in different units. Since atmospheric pressure calculations in our exercise use altitude in miles, converting feet to miles is necessary for accurate computation. To convert feet to miles, use the conversion factor:\[1 \text{ mile} = 5280 \text{ feet}\]For Asheville, North Carolina, and Phoenix, Arizona, we need to convert their altitudes given in feet into miles:

• Asheville: Altitude is given as 1985 feet. Convert this by \( \frac{1985}{5280} \approx 0.3765 \) miles.
• Phoenix: Altitude is provided as 1090 feet, which converts to \( \frac{1090}{5280} \approx 0.2068 \) miles.

Making accurate conversions ensures the precision of our calculations for atmospheric pressure.

Pressure Units Calculation

When calculating atmospheric pressure at a given altitude, it’s crucial to express the final result in the correct units. In our exercise, the units are pounds per square inch (psi). After using the equation \( P(a) = 14.7 \times e^{-0.21a} \) to find the atmospheric pressure, we express each answer to the nearest tenth of a psi.Evaluating our exponential expression and converting it into psi, we achieve the following results:

• For Mount McKinley at 3.85 miles, the result is approximately 6.8 psi.
• In Denver, at an altitude of 1 mile, the pressure is about 12.1 psi.
• In Asheville, with 0.3765 miles of altitude, the pressure computes to roughly 13.6 psi.
• For Phoenix, at 0.2068 miles, the atmospheric pressure is around 14.1 psi.

Presenting results in psi allows for consistent interpretation and comparison across different locations.